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, area,

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). Th ...

, 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 mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

,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 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 of the region in the plane that is bounded by the graph 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, a function whose derivative is the given function. In this case, they are called indefinite integrals. The fundamental theorem of calculus 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, 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 g ...

and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of

width. Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century, Henri Lebesgue generalized Riemann's formulation by introducing what is now referred to as the Lebesgue integral; 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 over which the integration is performed. For example, a line integral 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 in three-dimensional space.

History

Pre-calculus integration

The first documented systematic technique capable of determining integrals is the method of exhaustion 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 p ...

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 in the 3rd century BC and used to calculate the area of a circle, 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

of a sphere, area of an

ellipse In mathematics, an ellipse is a plane curve surrounding two 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 type of ellipse i ...

, the area under a parabola, the volume of a segment of a paraboloid 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 around the 3rd century AD by Liu Hui, 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 and Zu Geng to find the volume of a sphere. In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen ( AD) derived a formula for the sum of fourth powers. 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. The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of

Cavalieri Cavalieri is an Italian surname. Notable people with the surname include: * Bonaventura Cavalieri (1598–1647), Italian mathematician * Caterina Cavalieri (1755–1801), Austrian opera soprano * Diego Cavalieri (born 1982), Brazilian footb ...

with his method of Indivisibles, and work by Fermat, 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 Barrow may refer to: Places England * Barrow-in-Furness, Cumbria ** Borough of Barrow-in-Furness, local authority encompassing the wider area ** Barrow and Furness (UK Parliament constituency) * Barrow, Cheshire * Barrow, Gloucestershire * Barro ...

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. 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 by Leibniz 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

, 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. Bishop Berkeley memorably attacked the vanishing increments used by Newton, calling them " ghosts of departed quantities". Calculus acquired a firmer footing with the development of limits. Integration was first rigorously formalized, using limits, by Riemann. 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 Joseph ...

—to which Riemann's definition does not apply, and Lebesgue 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 sim ...

(a subfield of real analysis). 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 of an infinite Riemann sum, based on the hyperreal number system.

Historical notation

The notation for the indefinite integral was introduced by Gottfried Wilhelm Leibniz in 1675. He adapted the integral symbol, ∫, from the letter ''ſ'' (

long s The long s , also known as the medial s or initial s, is an 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" and "poſ� ...

), 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.

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 in 1690: "Ergo et horum Integralia aequantur".

Terminology and notation

In general, the integral of a real-valued function 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) whose derivative is the integrand. The fundamental theorem of calculus 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

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 sums of functions with respect to ''tagged partitions'' of an interval. A tagged partition of a closed interval 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 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...

/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 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 ...

, 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 functio ...

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 introduced the integral bearing his name, explaining this integral thus in a letter to Paul Montel: 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, 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 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

, which is defined by Darboux sums (restricted Riemann sums) yet is equivalent to the Riemann integral. 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, 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, which generalizes both the Riemann–Stieltjes and Lebesgue integrals. * The Daniell integral, which subsumes the Lebesgue integral and Lebesgue–Stieltjes integral without depending on measures. * 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 in 1933. * The Henstock–Kurzweil integral, variously defined by Arnaud Denjoy,

Oskar Perron Oskar Perron (7 May 1880 – 22 February 1975) was a German mathematician. He was a professor at the University of Heidelberg from 1914 to 1922 and at the University of Munich from 1922 to 1951. He made numerous contributions to differential ...

, 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 Rieman ...

, and developed by Ralph Henstock. * The Itô integral and Stratonovich integral, which define integration with respect to semimartingales such as Brownian motion. * 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 integral, which is defined for functions equipped with some additional "rough path" structure and generalizes stochastic integration against both semimartingales and processes such as the fractional Brownian motion. * The Choquet integral, a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953. * The Bochner integral, an extension of the Lebesgue integral to a more general class of functions, namely, those with a domain that is a Banach space.

Properties

Linearity

The collection of Riemann-integrable functions on a closed interval forms a vector space under the operations of pointwise addition and multiplication by a scalar, and the operation of integration :

f \mapsto \int_a^b f(x) \; dx

is a linear functional 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-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

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 topological vector space over a locally compact topological field . 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 numbers, 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 Daniell is a surname. Notable people with the surname include: * Alexander Daniell (1599–1668), Cornish landowner * Alfred Daniell (1853–1937) * Ave Daniell (1914–1999), American (gridiron) footballer * Charles Daniell (1827–1889), Major-G ...

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 Boundedness or bounded may refer to: Economics * Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision * Bounded e ...

interval and can be generalized to other notions of integral (Lebesgue and Daniell). * ''Upper and lower bounds.'' An integrable function on , is necessarily

on that interval. Thus there are real numbers 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 products and powers, and absolute values:

(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, plays a prominent role in

theory, where the left hand side is interpreted as the inner product of two square-integrable 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 In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of spaces. :Theorem (Hölder's inequality). Let be a measure space and let with . ...

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 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 L^p spaces.

Conventions

In this section, is a real-valued Riemann-integrable function. 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 , ...

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, should be zero. 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 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 . 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 [] that admits an antiderivative 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

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of a sequence of proper Riemann integrals 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

of proper integrals as one endpoint of the interval of integration approaches either a specified real number, 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 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

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 of two intervals

R=,b times,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 can be defined using Riemann sum 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 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 to be integrated is evaluated along a curve. 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 of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulas in physics have natural continuous analogs in terms of line integrals; for example, the fact that work 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 or gravitational field, 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 (which may be a curved set in space); it can be thought of as the Multiple integral, double integral analog of the line integral. The function to be integrated may be a

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 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 of with the unit Normal (geometry), surface normal 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 of electromagnetism.

Contour integrals

In complex analysis, the integrand is a complex-valued function 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.

Integrals of differential forms

A differential form is a mathematical concept in the fields of multivariable calculus, differential topology, and tensors. 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, which is similar to the cross product 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 plays the role of the gradient and Curl (mathematics), curl of vector calculus, and Stokes' theorem simultaneously generalizes the three theorems of vector calculus: the divergence theorem, Green's theorem, and the Kelvin-Stokes theorem.

Summations

The discrete equivalent of integration is summation. Summations and integrals can be put on the same foundations using the theory of Lebesgue integrals or time-scale calculus.

Functional integrals

An integration that is performed not over a variable (or, in physics, over a space or time dimension), but over a Function space, space of functions, is referred to as a functional integral.

Applications

Integrals are used extensively in many areas. For example, in probability theory, integrals are used to determine the probability of some random variable falling within a certain range. Moreover, the integral under an entire probability density function must equal 1, which provides a test of whether a function with no negative values could be a density function or not. Integrals can be used for computing the area of a two-dimensional region that has a curved boundary, as well as Volume integral, 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 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 to find quantities like Displacement (vector), displacement, time, and velocity. For example, in rectilinear motion, the displacement of an object over the time interval

[a,b]

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, where thermodynamic integration 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. 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 Mathematical singularity, 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, integration by parts, Trigonometric substitution, integration by trigonometric substitution, and Partial fractions in integration, integration by partial fractions. Alternative methods exist to compute more complex integrals. Many nonelementary integrals can be expanded in a Taylor series and integrated term by term. Occasionally, the resulting infinite series can be summed analytically. The method of convolution using Meijer G-functions 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 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. Computations of volumes of solid of revolution, solids of revolution can usually be done with disk integration or shell integration. Specific results which have been worked out by various techniques are collected in the Lists of integrals, list of integrals.

Symbolic

Many problems in mathematics, physics, and engineering involve integration where an explicit formula for the integral is desired. Extensive Lists of integrals, 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 systems 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 and Maple (software), Maple. 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 expression, closed form involving only elementary functions, include rational function, rational and exponential function, exponential functions, logarithm, trigonometric functions and inverse trigonometric functions, and the operations of multiplication and composition. The Risch algorithm 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, Maple (software), Maple and other computer algebra systems, does just that for functions and antiderivatives built from rational functions, Nth root, 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 (like the Legendre functions, the hypergeometric function, the gamma function, the incomplete gamma function 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 function, ''D''-finite functions, which are the solutions of linear differential equations 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. The rectangle method 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, 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 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. 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. One solution to this problem is Clenshaw–Curtis quadrature, in which the integrand is approximated by expanding it in terms of Chebyshev polynomials. Romberg's method 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 Interpolation, interpolate a polynomial through the approximations, and extrapolate to . Gaussian quadrature evaluates the function at the roots of a set of orthogonal polynomials. 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.

Mechanical

The area of an arbitrary two-dimensional shape can be determined using a measuring instrument called planimeter. The volume of irregular objects can be measured with precision by the fluid displacement (fluid), displaced as the object is submerged.

Geometrical

Area can sometimes be found via geometrical compass-and-straightedge constructions of an equivalent square.

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 and the partial derivative operator

\partial_x

. This can also be applied to functional integrals, allowing them to be computed by functional differentiation..

Examples

Using the Fundamental Theorem of Calculus

The fundamental theorem of calculus allows for straightforward calculations of basic functions.

\int_^ \sin(x)dx = -\cos(x) \big, ^_ = -\cos(\pi) - (-\cos(0)) = 2

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.

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. * 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 Integrals, Functions and mappings Linear operators in calculus