HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an integral assigns numbers to functions in a way that describes
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
,
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...
,
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
, and other concepts that arise by combining
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...
of the region in the plane that is bounded by the
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
of a given function between two points in the
real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
, a function whose derivative is the given function. In this case, they are called indefinite integrals. The
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
relates definite integrals with differentiation and provides a method to compute the definite integral of a function when its antiderivative is known. Although methods of calculating areas and volumes dated from
ancient Greek mathematics Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...
, the principles of integration were formulated independently by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
and
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
width.
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a
curvilinear In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is invertible, l ...
region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century,
Henri Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
generalized Riemann's formulation by introducing what is now referred to as the
Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
; it is more robust than Riemann's in the sense that a wider class of functions are Lebesgue-integrable. Integrals may be generalized depending on the type of the function as well as the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
over which the integration is performed. For example, a
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alt ...
is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting the two endpoints of the interval. In a
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may ...
, the curve is replaced by a piece of a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
in
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
.


History


Pre-calculus integration

The first documented systematic technique capable of determining integrals is the
method of exhaustion The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area bet ...
of the
ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...
astronomer Eudoxus (''ca.'' 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of divisions for which the area or volume was known. This method was further developed and employed by
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
in the 3rd century BC and used to calculate the
area of a circle In geometry, the area enclosed by a circle of radius is . Here the Greek letter represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159. One method of deriving this formula, which origi ...
, the
surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...
and
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
of a
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, area of an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
, the area under a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
, the volume of a segment of a
paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plane ...
of revolution, the volume of a segment of a
hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by defo ...
of revolution, and the area of a
spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:China China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's most populous country, with a population exceeding 1.4 billion, slightly ahead of India. China spans the equivalent of five time zones and ...
around the 3rd century AD by
Liu Hui Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu (The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state o ...
, who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians
Zu Chongzhi Zu Chongzhi (; 429–500 AD), courtesy name Wenyuan (), was a Chinese astronomer, mathematician, politician, inventor, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3 ...
and Zu Geng to find the volume of a sphere. In the Middle East, Hasan Ibn al-Haytham, Latinized as
Alhazen Ḥasan Ibn al-Haytham, Latinized as Alhazen (; full name ; ), was a medieval mathematician, astronomer, and physicist of the Islamic Golden Age from present-day Iraq.For the description of his main fields, see e.g. ("He is one of the prin ...
( AD) derived a formula for the sum of
fourth power In arithmetic and algebra, the fourth power of a number ''n'' is the result of multiplying four instances of ''n'' together. So: :''n''4 = ''n'' × ''n'' × ''n'' × ''n'' Fourth powers are also formed by multiplying a number by its cube. Further ...
s. He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a
paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plane ...
. The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of Cavalieri with his
method of Indivisibles In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that p ...
, and work by
Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he i ...
, began to lay the foundations of modern calculus, with Cavalieri computing the integrals of up to degree in Cavalieri's quadrature formula. Further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
. Wallis generalized Cavalieri's method, computing integrals of to a general power, including negative powers and fractional powers.


Leibniz and Newton

The major advance in integration came in the 17th century with the independent discovery of the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
by
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
and
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * ''Newton'' (film), a 2017 Indian film * Newton ( ...
. The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Leibniz and Newton developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions within continuous domains. This framework eventually became modern
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, whose notation for integrals is drawn directly from the work of Leibniz.


Formalization

While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of
rigour Rigour (British English) or rigor (American English; American and British English spelling differences#-our, -or, see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, su ...
.
Bishop Berkeley George Berkeley (; 12 March 168514 January 1753) – known as Bishop Berkeley (Bishop of Cloyne of the Anglican Church of Ireland) – was an Anglo-Irish philosopher whose primary achievement was the advancement of a theory he called "immateri ...
memorably attacked the vanishing increments used by Newton, calling them " ghosts of departed quantities". Calculus acquired a firmer footing with the development of
limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
. Integration was first rigorously formalized, using limits, by
Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
. Although all bounded piecewise continuous functions are Riemann-integrable on a bounded interval, subsequently more general functions were considered—particularly in the context of
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...
—to which Riemann's definition does not apply, and
Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
formulated a different definition of integral, founded in
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
(a subfield of
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include converg ...
). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. These approaches based on the real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the
standard part In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every suc ...
of an infinite Riemann sum, based on the
hyperreal number In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
system.


Historical notation

The notation for the indefinite integral was introduced by
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
in 1675. He adapted the
integral symbol The integral symbol: : (Unicode), \displaystyle \int (LaTeX) is used to denote integrals and antiderivatives in mathematics, especially in calculus. History The notation was introduced by the German mathematician Gottfried Wilhelm Leibniz in 1 ...
, ∫, from the letter ''ſ'' (
long s The long s , also known as the medial s or initial s, is an archaism, archaic form of the lowercase letter . It replaced the single ''s'', or one or both of the letters ''s'' in a 'double ''s sequence (e.g., "ſinfulneſs" for "sinfulness" ...
), standing for ''summa'' (written as ''ſumma''; Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by
Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French people, French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier an ...
in ''Mémoires'' of the French Academy around 1819–20, reprinted in his book of 1822.
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with or , which are used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.


First use of the term

The term was first printed in Latin by
Jacob Bernoulli Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Le ...
in 1690: "Ergo et horum Integralia aequantur".


Terminology and notation

In general, the integral of a
real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real fun ...
with respect to a real variable on an interval is written as :\int_^ f(x) \,dx. The integral sign represents integration. The symbol , called the differential of the variable , indicates that the variable of integration is . The function is called the integrand, the points and are called the limits (or bounds) of integration, and the integral is said to be over the interval , called the interval of integration.. A function is said to be if its integral over its domain is finite. If limits are specified, the integral is called a definite integral. When the limits are omitted, as in : \int f(x) \,dx, the integral is called an indefinite integral, which represents a class of functions (the
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
) whose derivative is the integrand. The
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
relates the evaluation of definite integrals to indefinite integrals. There are several extensions of the notation for integrals to encompass integration on unbounded domains and/or in multiple dimensions (see later sections of this article). In advanced settings, it is not uncommon to leave out when only the simple Riemann integral is being used, or the exact type of integral is immaterial. For instance, one might write \int_a^b (c_1f+c_2g) = c_1\int_a^b f + c_2\int_a^b g to express the linearity of the integral, a property shared by the Riemann integral and all generalizations thereof.


Interpretations

Integrals appear in many practical situations. For instance, from the length, width and depth of a swimming pool which is rectangular with a flat bottom, one can determine the volume of water it can contain, the area of its surface, and the length of its edge. But if it is oval with a rounded bottom, integrals are required to find exact and rigorous values for these quantities. In each case, one may divide the sought quantity into infinitely many
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
pieces, then sum the pieces to achieve an accurate approximation. For example, to find the area of the region bounded by the graph of the function between and , one can cross the interval in five steps (), then fill a rectangle using the right end height of each piece (thus ) and sum their areas to get an approximation of :\textstyle \sqrt\left(\frac-0\right)+\sqrt\left(\frac-\frac\right)+\cdots+\sqrt\left(\frac-\frac\right)\approx 0.7497, which is larger than the exact value. Alternatively, when replacing these subintervals by ones with the left end height of each piece, the approximation one gets is too low: with twelve such subintervals the approximated area is only 0.6203. However, when the number of pieces increase to infinity, it will reach a limit which is the exact value of the area sought (in this case, ). One writes :\int_^ \sqrt \,dx = \frac, which means is the result of a weighted sum of function values, , multiplied by the infinitesimal step widths, denoted by , on the interval .


Formal definitions

There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but also occasionally for pedagogical reasons. The most commonly used definitions are Riemann integrals and Lebesgue integrals.


Riemann integral

The Riemann integral is defined in terms of
Riemann sum In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
s of functions with respect to ''tagged partitions'' of an interval. A tagged partition of a
closed interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
on the real line is a finite sequence : a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_ \le t_n \le x_n = b . \,\! This partitions the interval into sub-intervals indexed by , each of which is "tagged" with a distinguished point . A ''Riemann sum'' of a function with respect to such a tagged partition is defined as : \sum_^n f(t_i) \, \Delta_i ; thus each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the width of sub-interval, . The ''mesh'' of such a tagged partition is the width of the largest sub-interval formed by the partition, . The ''Riemann integral'' of a function over the interval is equal to if: : For all \varepsilon > 0 there exists \delta > 0 such that, for any tagged partition , b/math> with mesh less than \delta, : \left, S - \sum_^n f(t_i) \, \Delta_i \ < \varepsilon. When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum becomes an upper (respectively, lower) Darboux sum, suggesting the close connection between the Riemann integral and the
Darboux integral In the branch of mathematics known as real analysis, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a functi ...
.


Lebesgue integral

It is often of interest, both in theory and applications, to be able to pass to the limit under the integral. For instance, a sequence of functions can frequently be constructed that approximate, in a suitable sense, the solution to a problem. Then the integral of the solution function should be the limit of the integrals of the approximations. However, many functions that can be obtained as limits are not Riemann-integrable, and so such limit theorems do not hold with the Riemann integral. Therefore, it is of great importance to have a definition of the integral that allows a wider class of functions to be integrated. Such an integral is the Lebesgue integral, that exploits the following fact to enlarge the class of integrable functions: if the values of a function are rearranged over the domain, the integral of a function should remain the same. Thus
Henri Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
introduced the integral bearing his name, explaining this integral thus in a letter to
Paul Montel Paul Antoine Aristide Montel (29 April 1876 – 22 January 1975) was a French mathematician. He was born in Nice, France and died in Paris, France. He researched mostly on holomorphic functions in complex analysis. Montel was a student of Émile ...
: As Folland puts it, "To compute the Riemann integral of , one partitions the domain into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of ". The definition of the Lebesgue integral thus begins with a
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
, μ. In the simplest case, the
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
of an interval is its width, , so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist. In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals. Using the "partitioning the range of " philosophy, the integral of a non-negative function should be the sum over of the areas between a thin horizontal strip between and . This area is just . Let . The Lebesgue integral of is then defined by : \int f = \int_0^\infty f^*(t)\,dt where the integral on the right is an ordinary improper Riemann integral ( is a strictly decreasing positive function, and therefore has a
well-defined In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A funct ...
improper Riemann integral). For a suitable class of functions (the
measurable function In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in di ...
s) this defines the Lebesgue integral. A general measurable function is Lebesgue-integrable if the sum of the absolute values of the areas of the regions between the graph of and the -axis is finite: : \int_E , f, \,d\mu < + \infty. In that case, the integral is, as in the Riemannian case, the difference between the area above the -axis and the area below the -axis:. : \int_E f \,d\mu = \int_E f^+ \,d\mu - \int_E f^- \,d\mu where : \begin & f^+(x) &&= \max \ &&= \begin f(x), & \text f(x) > 0, \\ 0, & \text \end\\ & f^-(x) &&= \max \ &&= \begin -f(x), & \text f(x) < 0, \\ 0, & \text \end \end


Other integrals

Although the Riemann and Lebesgue integrals are the most widely used definitions of the integral, a number of others exist, including: * The
Darboux integral In the branch of mathematics known as real analysis, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a functi ...
, which is defined by Darboux sums (restricted Riemann sums) yet is equivalent to the
Riemann integral In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
. A function is Darboux-integrable if and only if it is Riemann-integrable. Darboux integrals have the advantage of being easier to define than Riemann integrals. * The
Riemann–Stieltjes integral In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an inst ...
, an extension of the Riemann integral which integrates with respect to a function as opposed to a variable. * The Lebesgue–Stieltjes integral, further developed by
Johann Radon Johann Karl August Radon (; 16 December 1887 – 25 May 1956) was an Austrian mathematician. His doctoral dissertation was on the calculus of variations (in 1910, at the University of Vienna). Life RadonBrigitte Bukovics: ''Biography of Johan ...
, which generalizes both the Riemann–Stieltjes and Lebesgue integrals. * The
Daniell integral In mathematics, the Daniell integral is a type of integration that generalizes the concept of more elementary versions such as the Riemann integral to which students are typically first introduced. One of the main difficulties with the traditional f ...
, which subsumes the Lebesgue integral and Lebesgue–Stieltjes integral without depending on
measures Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Measu ...
. * The
Haar integral In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This Measure (mathematics), measure was introduced by Alfré ...
, used for integration on locally compact topological groups, introduced by
Alfréd Haar Alfréd Haar ( hu, Haar Alfréd; 11 October 1885, Budapest – 16 March 1933, Szeged) was a Hungarian mathematician. In 1904 he began to study at the University of Göttingen. His doctorate was supervised by David Hilbert. The Haar me ...
in 1933. * The
Henstock–Kurzweil integral In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced ), Luzin integral or Perron integral, but not to be confused with the more general wide ...
, variously defined by
Arnaud Denjoy Arnaud Denjoy (; 5 January 1884 – 21 January 1974) was a French mathematician. Biography Denjoy was born in Auch, Gers. His contributions include work in harmonic analysis and differential equations. Henstock–Kurzweil integral, His integral ...
, Oskar Perron, and (most elegantly, as the gauge integral)
Jaroslav Kurzweil Jaroslav Kurzweil (, 7 May 1926, Prague – 17 March 2022) was a Czech mathematician. Biography Born in Prague, Czechoslovakia, he was a specialist in ordinary differential equations and defined the Henstock–Kurzweil integral in terms of Riema ...
, and developed by
Ralph Henstock Ralph Henstock (2 June 1923 – 17 January 2007) was an English mathematician and author. As an Integral#Other integrals, Integration theorist, he is notable for Henstock–Kurzweil integral. Henstock brought the theory to a highly developed stage ...
. * The Itô integral and
Stratonovich integral In stochastic processes, the Stratonovich integral (developed simultaneously by Ruslan Stratonovich and Donald Fisk) is a stochastic integral, the most common alternative to the Itô integral. Although the Itô integral is the usual choice in a ...
, which define integration with respect to
semimartingale In probability theory, a real valued stochastic process ''X'' is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the l ...
s such as
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
. * The
Young integral Young may refer to: * Offspring, the product of reproduction of a new organism produced by one or more parents * Youth, the time of life when one is young, often meaning the time between childhood and adulthood Music * The Young, an American roc ...
, which is a kind of Riemann–Stieltjes integral with respect to certain functions of unbounded variation. * The
rough path In stochastic analysis, a rough path is a generalization of the notion of smooth path allowing to construct a robust solution theory for controlled differential equations driven by classically irregular signals, for example a Wiener process. The the ...
integral, which is defined for functions equipped with some additional "rough path" structure and generalizes stochastic integration against both
semimartingale In probability theory, a real valued stochastic process ''X'' is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the l ...
s and processes such as the
fractional Brownian motion In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a continuous-time Gauss ...
. * The
Choquet integral A Choquet integral is a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953. It was initially used in statistical mechanics and potential theory, but found its way into decision theory in the 1980s, wher ...
, a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953. * The
Bochner integral In mathematics, the Bochner integral, named for Salomon Bochner, extends the definition of Lebesgue integral to functions that take values in a Banach space, as the limit of integrals of simple functions. Definition Let (X, \Sigma, \mu) be a meas ...
, an extension of the Lebesgue integral to a more general class of functions, namely, those with a domain that is a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
.


Properties


Linearity

The collection of Riemann-integrable functions on a closed interval forms a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
under the operations of
pointwise addition In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
and multiplication by a scalar, and the operation of integration : f \mapsto \int_a^b f(x) \; dx is a
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the s ...
on this vector space. Thus, the collection of integrable functions is closed under taking linear combinations, and the integral of a linear combination is the linear combination of the integrals:. : \int_a^b (\alpha f + \beta g)(x) \, dx = \alpha \int_a^b f(x) \,dx + \beta \int_a^b g(x) \, dx. \, Similarly, the set of
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
-valued Lebesgue-integrable functions on a given
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
with measure is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral : f\mapsto \int_E f \, d\mu is a linear functional on this vector space, so that: : \int_E (\alpha f + \beta g) \, d\mu = \alpha \int_E f \, d\mu + \beta \int_E g \, d\mu. More generally, consider the vector space of all
measurable function In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in di ...
s on a measure space , taking values in a
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
over a locally compact
topological field In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is w ...
. Then one may define an abstract integration map assigning to each function an element of or the symbol , : f\mapsto\int_E f \,d\mu, \, that is compatible with linear combinations. In this situation, the linearity holds for the subspace of functions whose integral is an element of (i.e. "finite"). The most important special cases arise when is , , or a finite extension of the field of
p-adic number In mathematics, the -adic number system for any prime number  extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensi ...
s, and is a finite-dimensional vector space over , and when and is a complex
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. Linearity, together with some natural continuity properties and normalization for a certain class of "simple" functions, may be used to give an alternative definition of the integral. This is the approach of Daniell for the case of real-valued functions on a set , generalized by
Nicolas Bourbaki Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (Paris), École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally in ...
to functions with values in a locally compact topological vector space. See for an axiomatic characterization of the integral.


Inequalities

A number of general inequalities hold for Riemann-integrable functions defined on a closed and bounded interval and can be generalized to other notions of integral (Lebesgue and Daniell). * ''Upper and lower bounds.'' An integrable function on , is necessarily bounded on that interval. Thus there are
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s and so that for all in . Since the lower and upper sums of over are therefore bounded by, respectively, and , it follows that m(b - a) \leq \int_a^b f(x) \, dx \leq M(b - a). * ''Inequalities between functions.'' If for each in then each of the upper and lower sums of is bounded above by the upper and lower sums, respectively, of . Thus \int_a^b f(x) \, dx \leq \int_a^b g(x) \, dx. This is a generalization of the above inequalities, as is the integral of the constant function with value over . In addition, if the inequality between functions is strict, then the inequality between integrals is also strict. That is, if for each in , then \int_a^b f(x) \, dx < \int_a^b g(x) \, dx. * ''Subintervals.'' If is a subinterval of and is non-negative for all , then \int_c^d f(x) \, dx \leq \int_a^b f(x) \, dx. * ''Products and absolute values of functions.'' If and are two functions, then we may consider their
pointwise product In mathematics, the pointwise product of two functions is another function, obtained by multiplying the images of the two functions at each value in the domain. If and are both functions with domain and codomain , and elements of can be mul ...
s and powers, and
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
s: (fg)(x)= f(x) g(x), \; f^2 (x) = (f(x))^2, \; , f, (x) = , f(x), . If is Riemann-integrable on then the same is true for , and \left, \int_a^b f(x) \, dx \ \leq \int_a^b , f(x) , \, dx. Moreover, if and are both Riemann-integrable then is also Riemann-integrable, and \left( \int_a^b (fg)(x) \, dx \right)^2 \leq \left( \int_a^b f(x)^2 \, dx \right) \left( \int_a^b g(x)^2 \, dx \right). This inequality, known as the
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality fo ...
, plays a prominent role in
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
theory, where the left hand side is interpreted as the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
of two
square-integrable In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real number, real- or complex number, complex-valued measurable function for which the integral of the s ...
functions and on the interval . * ''Hölder's inequality''.. Suppose that and are two real numbers, with , and and are two Riemann-integrable functions. Then the functions and are also integrable and the following Hölder's inequality holds: \left, \int f(x)g(x)\,dx\ \leq \left(\int \left, f(x)\^p\,dx \right)^ \left(\int\left, g(x)\^q\,dx\right)^. For , Hölder's inequality becomes the Cauchy–Schwarz inequality. * ''Minkowski inequality''. Suppose that is a real number and and are Riemann-integrable functions. Then and are also Riemann-integrable and the following
Minkowski inequality In mathematical analysis, the Minkowski inequality establishes that the L''p'' spaces are normed vector spaces. Let ''S'' be a measure space, let and let ''f'' and ''g'' be elements of L''p''(''S''). Then is in L''p''(''S''), and we have the t ...
holds: \left(\int \left, f(x)+g(x)\^p\,dx \right)^ \leq \left(\int \left, f(x)\^p\,dx \right)^ + \left(\int \left, g(x)\^p\,dx \right)^. An analogue of this inequality for Lebesgue integral is used in construction of Lp spaces.


Conventions

In this section, is a
real-valued In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an i ...
Riemann-integrable
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
. The integral : \int_a^b f(x) \, dx over an interval is defined if . This means that the upper and lower sums of the function are evaluated on a partition whose values are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating within intervals where an interval with a higher index lies to the right of one with a lower index. The values and , the end-points of the interval, are called the
limits of integration In calculus and mathematical analysis the limits of integration (or bounds of integration) of the integral : \int_a^b f(x) \, dx of a Riemann integrable function f defined on a closed and bounded interval are the real numbers a and b , in w ...
of . Integrals can also be defined if :'''' :\int_a^b f(x) \, dx = - \int_b^a f(x) \, dx. With , this implies: :\int_a^a f(x) \, dx = 0. The first convention is necessary in consideration of taking integrals over subintervals of ; the second says that an integral taken over a degenerate interval, or a
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...
, should be
zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
. One reason for the first convention is that the integrability of on an interval implies that is integrable on any subinterval , but in particular integrals have the property that if is any element of , then:'''' : \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx. With the first convention, the resulting relation : \begin \int_a^c f(x) \, dx &= \int_a^b f(x) \, dx - \int_c^b f(x) \, dx \\ & = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx \end is then well-defined for any cyclic permutation of , , and .


Fundamental theorem of calculus

The ''fundamental theorem of calculus'' is the statement that differentiation and integration are inverse operations: if a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
is first integrated and then differentiated, the original function is retrieved. An important consequence, sometimes called the ''second fundamental theorem of calculus'', allows one to compute integrals by using an antiderivative of the function to be integrated.


First theorem

Let be a continuous real-valued function defined on a
closed interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
. Let be the function defined, for all in , by : F(x) = \int_a^x f(t)\, dt. Then, is continuous on , differentiable on the open interval , and : F'(x) = f(x) for all in .


Second theorem

Let be a real-valued function defined on a
closed interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
[] that admits an
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
on . That is, and are functions such that for all in , : f(x) = F'(x). If is integrable on then : \int_a^b f(x)\,dx = F(b) - F(a).


Extensions


Improper integrals

A "proper" Riemann integral assumes the integrand is defined and finite on a closed and bounded interval, bracketed by the limits of integration. An improper integral occurs when one or more of these conditions is not satisfied. In some cases such integrals may be defined by considering the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of proper
Riemann integral In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
s on progressively larger intervals. If the interval is unbounded, for instance at its upper end, then the improper integral is the limit as that endpoint goes to infinity: : \int_a^\infty f(x)\,dx = \lim_ \int_a^b f(x)\,dx. If the integrand is only defined or finite on a half-open interval, for instance , then again a limit may provide a finite result: : \int_a^b f(x)\,dx = \lim_ \int_^ f(x)\,dx. That is, the improper integral is the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of proper integrals as one endpoint of the interval of integration approaches either a specified
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
, or , or . In more complicated cases, limits are required at both endpoints, or at interior points.


Multiple integration

Just as the definite integral of a positive function of one variable represents the
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...
of the region between the graph of the function and the ''x''-axis, the ''double integral'' of a positive function of two variables represents the
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
of the region between the surface defined by the function and the plane that contains its domain. For example, a function in two dimensions depends on two real variables, ''x'' and ''y'', and the integral of a function ''f'' over the rectangle ''R'' given as the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
of two intervals R= ,btimes ,d/math> can be written : \int_R f(x,y)\,dA where the differential indicates that integration is taken with respect to area. This
double integral In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ...
can be defined using
Riemann sum In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
s, and represents the (signed) volume under the graph of over the domain ''R''.. Under suitable conditions (e.g., if ''f'' is continuous),
Fubini's theorem In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...
states that this integral can be expressed as an equivalent iterated integral : \int_a^b\left int_c^d f(x,y)\,dy\right,dx. This reduces the problem of computing a double integral to computing one-dimensional integrals. Because of this, another notation for the integral over ''R'' uses a double integral sign: : \iint_R f(x,y) \, dA. Integration over more general domains is possible. The integral of a function ''f'', with respect to volume, over an ''n-''dimensional region ''D'' of \mathbb^n is denoted by symbols such as: : \int_D f(\mathbf x) d^n\mathbf x \ = \int_D f\,dV.


Line integrals and surface integrals

The concept of an integral can be extended to more general domains of integration, such as curved lines and surfaces inside higher-dimensional spaces. Such integrals are known as line integrals and surface integrals respectively. These have important applications in physics, as when dealing with vector fields. A ''line integral'' (sometimes called a ''path integral'') is an integral where the
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
to be integrated is evaluated along a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
. Various different line integrals are in use. In the case of a closed curve it is also called a ''contour integral''. The function to be integrated may be a
scalar field In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly
arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
or, for a vector field, the
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on
intervals Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets * A statistical level of measurement * Interval e ...
. Many simple formulas in physics have natural continuous analogs in terms of line integrals; for example, the fact that
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal t ...
is equal to
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
, , multiplied by displacement, , may be expressed (in terms of vector quantities) as: : W=\mathbf F\cdot\mathbf s. For an object moving along a path in a vector field such as an
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
or
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
, the total work done by the field on the object is obtained by summing up the differential work done in moving from to . This gives the line integral : W=\int_C \mathbf F\cdot d\mathbf s. A ''surface integral'' generalizes double integrals to integration over a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
(which may be a curved set in
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
); it can be thought of as the
double integral In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ...
analog of the
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alt ...
. The function to be integrated may be a
scalar field In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
or a vector field. The value of the surface integral is the sum of the field at all points on the surface. This can be achieved by splitting the surface into surface elements, which provide the partitioning for Riemann sums. For an example of applications of surface integrals, consider a vector field on a surface ; that is, for each point in , is a vector. Imagine that a fluid flows through , such that determines the velocity of the fluid at . The
flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
is defined as the quantity of fluid flowing through in unit amount of time. To find the flux, one need to take the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
of with the unit
surface normal In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
to at each point, which will give a scalar field, which is integrated over the surface: : \int_S \cdot \,d. The fluid flux in this example may be from a physical fluid such as water or air, or from electrical or magnetic flux. Thus surface integrals have applications in physics, particularly with the
classical theory Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
of
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
.


Contour integrals

In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, the integrand is a
complex-valued function Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
of a complex variable instead of a real function of a real variable . When a complex function is integrated along a curve \gamma in the complex plane, the integral is denoted as follows : \int_\gamma f(z)\,dz. This is known as a
contour integral In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. ...
.


Integrals of differential forms

A
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
is a mathematical concept in the fields of
multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather th ...
,
differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
, and
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
s. Differential forms are organized by degree. For example, a one-form is a weighted sum of the differentials of the coordinates, such as: : E(x,y,z)\,dx + F(x,y,z)\,dy + G(x,y,z)\, dz where ''E'', ''F'', ''G'' are functions in three dimensions. A differential one-form can be integrated over an oriented path, and the resulting integral is just another way of writing a line integral. Here the basic differentials ''dx'', ''dy'', ''dz'' measure infinitesimal oriented lengths parallel to the three coordinate axes. A differential two-form is a sum of the form : G(x,y,z) \, dx\wedge dy + E(x,y,z) \, dy\wedge dz + F(x,y,z) \, dz\wedge dx. Here the basic two-forms dx\wedge dy, dz\wedge dx, dy\wedge dz measure oriented areas parallel to the coordinate two-planes. The symbol \wedge denotes the
wedge product A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converti ...
, which is similar to the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
in the sense that the wedge product of two forms representing oriented lengths represents an oriented area. A two-form can be integrated over an oriented surface, and the resulting integral is equivalent to the surface integral giving the flux of E\mathbf i+F\mathbf j+G\mathbf k. Unlike the cross product, and the three-dimensional vector calculus, the wedge product and the calculus of differential forms makes sense in arbitrary dimension and on more general manifolds (curves, surfaces, and their higher-dimensional analogs). The
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
plays the role of the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
and
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fi ...
of vector calculus, and
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
simultaneously generalizes the three theorems of vector calculus: the
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
,
Green's theorem In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem. Theorem Let be a positively orient ...
, and the
Kelvin-Stokes theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
.


Summations

The discrete equivalent of integration is
summation In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, mat ...
. Summations and integrals can be put on the same foundations using the theory of
Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
s or
time-scale calculus In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying ...
.


Functional integrals

An integration that is performed not over a variable (or, in physics, over a space or time dimension), but over a
space of functions In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vector ...
, is referred to as a functional integral.


Applications

Integrals are used extensively in many areas. For example, in
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, integrals are used to determine the probability of some
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
falling within a certain range. Moreover, the integral under an entire
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
must equal 1, which provides a test of whether a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
with no negative values could be a density function or not. Integrals can be used for computing the
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...
of a two-dimensional region that has a curved boundary, as well as computing the volume of a three-dimensional object that has a curved boundary. The area of a two-dimensional region can be calculated using the aforementioned definite integral. The volume of a three-dimensional object such as a disc or washer can be computed by
disc integration Disc integration, also known in integral calculus as the disc method, is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel" to the axis of revolution. This method mod ...
using the equation for the volume of a cylinder, \pi r^2 h , where r is the radius. In the case of a simple disc created by rotating a curve about the -axis, the radius is given by , and its height is the differential . Using an integral with bounds and , the volume of the disc is equal to:\pi \int_a^b f^2 (x) \, dx.Integrals are also used in physics, in areas like
kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
to find quantities like
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
,
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
, and
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
. For example, in rectilinear motion, the displacement of an object over the time interval ,b/math> is given by: : x(b)-x(a) = \int_a^b v(t) \,dt, where v(t) is the velocity expressed as a function of time. The work done by a force F(x) (given as a function of position) from an initial position A to a final position B is: : W_ = \int_A^B F(x)\,dx. Integrals are also used in
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
, where
thermodynamic integration Thermodynamic integration is a method used to compare the difference in free energy between two given states (e.g., A and B) whose potential energies U_A and U_B have different dependences on the spatial coordinates. Because the free energy of a ...
is used to calculate the difference in free energy between two given states.


Computation


Analytical

The most basic technique for computing definite integrals of one real variable is based on the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
. Let be the function of to be integrated over a given interval . Then, find an antiderivative of ; that is, a function such that on the interval. Provided the integrand and integral have no singularities on the path of integration, by the fundamental theorem of calculus, :\int_a^b f(x)\,dx=F(b)-F(a). Sometimes it is necessary to use one of the many techniques that have been developed to evaluate integrals. Most of these techniques rewrite one integral as a different one which is hopefully more tractable. Techniques include
integration by substitution In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can ...
,
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
, integration by trigonometric substitution, and
integration by partial fractions In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...
. Alternative methods exist to compute more complex integrals. Many
nonelementary integral In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an ''elementary function'' (i.e. a function constructed from a finite number of quotients of constan ...
s can be expanded in a
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
and integrated term by term. Occasionally, the resulting infinite series can be summed analytically. The method of convolution using
Meijer G-function In mathematics, the G-function was introduced by as a very general function intended to include most of the known special functions as particular cases. This was not the only attempt of its kind: the generalized hypergeometric function and the M ...
s can also be used, assuming that the integrand can be written as a product of Meijer G-functions. There are also many less common ways of calculating definite integrals; for instance,
Parseval's identity In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is a generalized Pythagorean theorem for inner-product spaces (which ...
can be used to transform an integral over a rectangular region into an infinite sum. Occasionally, an integral can be evaluated by a trick; for an example of this, see
Gaussian integral The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f(x) = e^ over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is \int_^\infty e^\,dx = \s ...
. Computations of volumes of
solids of revolution In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the ''axis of revolution'') that lies on the same plane. The surface created by this revolution and which bounds the solid is the ...
can usually be done with
disk integration Disc integration, also known in integral calculus as the disc method, is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel" to the axis of revolution. This method mod ...
or
shell integration Shell integration (the shell method in integral calculus) is a method for calculating the volume of a solid of revolution, when integrating along an axis ''perpendicular to'' the axis of revolution. This is in contrast to disc integration whi ...
. Specific results which have been worked out by various techniques are collected in the
list of integrals Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not ...
.


Symbolic

Many problems in mathematics, physics, and engineering involve integration where an explicit formula for the integral is desired. Extensive tables of integrals have been compiled and published over the years for this purpose. With the spread of computers, many professionals, educators, and students have turned to
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The de ...
s that are specifically designed to perform difficult or tedious tasks, including integration. Symbolic integration has been one of the motivations for the development of the first such systems, like
Macsyma Macsyma (; "Project MAC's SYmbolic MAnipulator") is one of the oldest general-purpose computer algebra systems still in wide use. It was originally developed from 1968 to 1982 at MIT's Project MAC. In 1982, Macsyma was licensed to Symbolics and ...
and
Maple ''Acer'' () is a genus of trees and shrubs commonly known as maples. The genus is placed in the family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated since http ...
. A major mathematical difficulty in symbolic integration is that in many cases, a relatively simple function does not have integrals that can be expressed in closed form involving only
elementary function In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponen ...
s, include
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
and
exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above * Exponential decay, decrease at a rate proportional to value *Exp ...
functions,
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
,
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
and
inverse trigonometric functions In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted Domain of a fu ...
, and the operations of multiplication and composition. The
Risch algorithm In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra ...
provides a general criterion to determine whether the antiderivative of an elementary function is elementary, and to compute it if it is. However, functions with closed expressions of antiderivatives are the exception, and consequently, computerized algebra systems have no hope of being able to find an antiderivative for a randomly constructed elementary function. On the positive side, if the 'building blocks' for antiderivatives are fixed in advance, it may still be possible to decide whether the antiderivative of a given function can be expressed using these blocks and operations of multiplication and composition, and to find the symbolic answer whenever it exists. The Risch algorithm, implemented in
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...
,
Maple ''Acer'' () is a genus of trees and shrubs commonly known as maples. The genus is placed in the family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated since http ...
and other
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The de ...
s, does just that for functions and antiderivatives built from rational functions, radicals, logarithm, and exponential functions. Some special integrands occur often enough to warrant special study. In particular, it may be useful to have, in the set of antiderivatives, the
special functions Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ...
(like the Legendre functions, the hypergeometric function, the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
, the
incomplete gamma function In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, which ...
and so on). Extending the Risch's algorithm to include such functions is possible but challenging and has been an active research subject. More recently a new approach has emerged, using ''D''-finite functions, which are the solutions of
linear differential equation In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b( ...
s with polynomial coefficients. Most of the elementary and special functions are ''D''-finite, and the integral of a ''D''-finite function is also a ''D''-finite function. This provides an algorithm to express the antiderivative of a ''D''-finite function as the solution of a differential equation. This theory also allows one to compute the definite integral of a ''D''-function as the sum of a series given by the first coefficients, and provides an algorithm to compute any coefficient.


Numerical

Definite integrals may be approximated using several methods of
numerical integration In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations ...
. The
rectangle method In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
relies on dividing the region under the function into a series of rectangles corresponding to function values and multiplies by the step width to find the sum. A better approach, the
trapezoidal rule In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule; see Trapezoid for more information on terminology) is a technique for approximating the definite integral. \int_a^b f(x) \, dx. The trapezoidal rule works b ...
, replaces the rectangles used in a Riemann sum with trapezoids. The trapezoidal rule weights the first and last values by one half, then multiplies by the step width to obtain a better approximation. The idea behind the trapezoidal rule, that more accurate approximations to the function yield better approximations to the integral, can be carried further:
Simpson's rule In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761). The most basic of these rules, called Simpson's 1/3 rule, or just Simpson's rule, reads \int_a^b f(x) \, ...
approximates the integrand by a piecewise quadratic function. Riemann sums, the trapezoidal rule, and Simpson's rule are examples of a family of quadrature rules called the
Newton–Cotes formulas In numerical analysis, the Newton–Cotes formulas, also called the Newton–Cotes quadrature rules or simply Newton–Cotes rules, are a group of formulas for numerical integration (also called ''quadrature'') based on evaluating the integrand at ...
. The degree Newton–Cotes quadrature rule approximates the polynomial on each subinterval by a degree ' polynomial. This polynomial is chosen to interpolate the values of the function on the interval. Higher degree Newton–Cotes approximations can be more accurate, but they require more function evaluations, and they can suffer from numerical inaccuracy due to
Runge's phenomenon In the mathematical field of numerical analysis, Runge's phenomenon () is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation ...
. One solution to this problem is
Clenshaw–Curtis quadrature Clenshaw–Curtis quadrature and Fejér quadrature are methods for numerical integration, or "quadrature", that are based on an expansion of the integrand in terms of Chebyshev polynomials. Equivalently, they employ a change of variables x = \cos ...
, in which the integrand is approximated by expanding it in terms of
Chebyshev polynomials The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshe ...
.
Romberg's method In numerical analysis, Romberg's method is used to estimate the definite integral \int_a^b f(x) \, dx by applying Richardson extrapolation repeatedly on the trapezium rule or the rectangle rule (midpoint rule). The estimates generate a triangul ...
halves the step widths incrementally, giving trapezoid approximations denoted by , , and so on, where is half of . For each new step size, only half the new function values need to be computed; the others carry over from the previous size. It then
interpolate In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a n ...
a polynomial through the approximations, and extrapolate to .
Gaussian quadrature In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more ...
evaluates the function at the roots of a set of
orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product. The most widely used orthogonal polynomial ...
. An -point Gaussian method is exact for polynomials of degree up to . The computation of higher-dimensional integrals (for example, volume calculations) makes important use of such alternatives as
Monte Carlo integration In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand a ...
.


Mechanical

The area of an arbitrary two-dimensional shape can be determined using a measuring instrument called
planimeter A planimeter, also known as a platometer, is a measuring instrument used to determine the area of an arbitrary two-dimensional shape. Construction There are several kinds of planimeters, but all operate in a similar way. The precise way in whic ...
. The volume of irregular objects can be measured with precision by the fluid displaced as the object is submerged.


Geometrical

Area can sometimes be found via
geometrical Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
compass-and-straightedge constructions of an equivalent
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
.


Integration by differentiation

Kempf, Jackson and Morales demonstrated mathematical relations that allow an integral to be calculated by means of differentiation. Their calculus involves the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
and the
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
operator \partial_x. This can also be applied to
functional integral Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions. Functional integrals arise in probability, in the study of partial differentia ...
s, allowing them to be computed by functional differentiation..


Examples


Using the Fundamental Theorem of Calculus

The
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
allows for straightforward calculations of basic functions. \int_^ \sin(x)dx = -\cos(x) \big, ^_ = -\cos(\pi) - (-\cos(0)) = 2


See also

* *


Notes


References


Bibliography

* * * . In particular chapters III and IV. * * * * * *
Available in translation as *
(Originally published by Cambridge University Press, 1897, based on J. L. Heiberg's Greek version.) * * * * * * * * * * * . * * . * *


External links

*
Online Integral Calculator
Wolfram Alpha WolframAlpha ( ) is an answer engine developed by Wolfram Research. It answers factual queries by computing answers from externally sourced data. WolframAlpha was released on May 18, 2009 and is based on Wolfram's earlier product Wolfram Mathe ...
.


Online books

* Keisler, H. Jerome
Elementary Calculus: An Approach Using Infinitesimals
University of Wisconsin * Stroyan, K. D.

University of Iowa * Mauch, Sean

CIT, an online textbook that includes a complete introduction to calculus * Crowell, Benjamin
''Calculus''
Fullerton College, an online textbook * Garrett, Paul
Notes on First-Year Calculus
* Hussain, Faraz
Understanding Calculus
an online textbook * Johnson, William Woolsey (1909
Elementary Treatise on Integral Calculus
link from
HathiTrust HathiTrust Digital Library is a large-scale collaborative repository of digital content from research libraries including content digitized via Google Books and the Internet Archive digitization initiatives, as well as content digitized locally ...
. * Kowalk, W. P.
''Integration Theory''
University of Oldenburg. A new concept to an old problem. Online textbook * Sloughter, Dan
Difference Equations to Differential Equations
an introduction to calculus

at ''Holistic Numerical Methods Institute'' * P. S. Wang
Evaluation of Definite Integrals by Symbolic Manipulation
(1972) — a cookbook of definite integral techniques {{Machine learning evaluation metrics Functions and mappings Linear operators in calculus