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In mathematics (specifically
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 ...
), a multiple integral is a definite integral of a
function of several real variables In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function o ...
, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number plane) are called double integrals, and integrals of a function of three variables over a region in \mathbb^3 (real-number 3D space) are called triple integrals. For multiple integrals of a single-variable function, see the
Cauchy formula for repeated integration The Cauchy formula for repeated integration, named after Augustin-Louis Cauchy, allows one to compress ''n'' antidifferentiations of a function into a single integral (cf. Cauchy's formula). Scalar case Let ''f'' be a continuous function on the r ...
.


Introduction

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 or planar lamina, while '' surface area'' refers to the area of an ope ...
of the region between the graph of the function and the -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). Th ...
of the region between the surface defined by the function (on the three-dimensional
Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
where ) and the plane which contains its domain. If there are more variables, a multiple integral will yield
hypervolume A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...
s of multidimensional functions. Multiple integration of a function in variables: over a domain is most commonly represented by nested integral signs in the reverse order of execution (the leftmost integral sign is computed last), followed by the function and integrand arguments in proper order (the integral with respect to the rightmost argument is computed last). The domain of integration is either represented symbolically for every argument over each integral sign, or is abbreviated by a variable at the rightmost integral sign: : \int \cdots \int_\mathbf\, f(x_1,x_2,\ldots,x_n) \,dx_1 \!\cdots dx_n Since 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 symbolicall ...
is only defined for functions of a single real variable, the usual definition of the
indefinite integral 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 symbolicall ...
does not immediately extend to the multiple integral.


Mathematical definition

For , consider a so-called "half-open" -dimensional hyperrectangular domain , defined as: :T=
Partition_ Partition__may_refer_to: _Computing__Hardware_ *__Disk_partitioning,_the_division_of_a_hard_disk_drive *__Memory_partition,_a_subdivision_of_a_computer's_memory,_usually_for_use_by_a_single_job __Software_ *_Partition_(database),_the_division_of_a_...
_each_interval__into_a_finite_family__of_non-overlapping_subintervals_,_with_each_subinterval_closed_at_the_left_end,_and_open_at_the_right_end. Then_the_finite_family_of_subrectangles__given_by :C=I_1\times_I_2\times_\cdots_\times_I_n is_a_partition_(set_theory).html" "title="partition_(set_theory).html" "title="a_1, b_1) \times [ a_2, b_2) \times \cdots \times [ a_n, b_n) \subseteq \R^n. partition (set theory)">Partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
each interval into a finite family of non-overlapping subintervals , with each subinterval closed at the left end, and open at the right end. Then the finite family of subrectangles given by :C=I_1\times I_2\times \cdots \times I_n is a partition (set theory)">partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
 of ; that is, the subrectangles are non-overlapping and their union is . Let be a function defined on . Consider a partition of as defined above, such that is a family of subrectangles and :T=C_1\cup C_2\cup \cdots \cup C_m We can approximate the total -dimensional volume bounded below by the -dimensional hyperrectangle and above by the -dimensional graph of with the following Riemann sum: :\sum_^m f(P_k)\, \operatorname(C_k) where is a point in and is the product of the lengths of the intervals whose Cartesian product is , also known as the measure of . The diameter of a subrectangle is the largest of the lengths of the intervals whose Cartesian product is . The diameter of a given partition of is defined as the largest of the diameters of the subrectangles in the partition. Intuitively, as the diameter of the partition is restricted smaller and smaller, the number of subrectangles gets larger, and the measure of each subrectangle grows smaller. The function is said to be Riemann integrable if the limit :S=\lim_ \sum_^m f(P_k)\, \operatorname (C_k) exists, where the limit is taken over all possible partitions of of diameter at most . If is Riemann integrable, is called the Riemann integral of over and is denoted : \int \cdots \int_T\, f(x_1,x_2,\ldots,x_n) \,dx_1 \!\cdots dx_n Frequently this notation is abbreviated as : \int_T\!f(\mathbf)\,d^n\mathbf. where represents the -tuple and is the -dimensional volume differential. The Riemann integral of a function defined over an arbitrary bounded -dimensional set can be defined by extending that function to a function defined over a half-open rectangle whose values are zero outside the domain of the original function. Then the integral of the original function over the original domain is defined to be the integral of the extended function over its rectangular domain, if it exists. In what follows the Riemann integral in dimensions will be called the multiple integral.


Properties

Multiple integrals have many properties common to those of integrals of functions of one variable (linearity, commutativity, monotonicity, and so on). One important property of multiple integrals is that the value of an integral is independent of the order of integrands under certain conditions. This property is popularly known as
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 ...
.


Particular cases

In the case of the integral : l = \iint_T f(x,y)\, dx\, dy is the double integral of on , and if T \subseteq \R^3 the integral : l = \iiint_T f(x,y,z)\, dx\, dy\, dz is the triple integral of on . Notice that, by convention, the double integral has two integral signs, and the triple integral has three; this is a notational convention which is convenient when computing a multiple integral as an iterated integral, as shown later in this article.


Methods of integration

The resolution of problems with multiple integrals consists, in most cases, of finding a way to reduce the multiple integral to an
iterated integral In multivariable calculus, an iterated integral is the result of applying integrals to a function of more than one variable (for example f(x,y) or f(x,y,z)) in a way that each of the integrals considers some of the variables as given constants. ...
, a series of integrals of one variable, each being directly solvable. For continuous functions, this is justified by
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 ...
. Sometimes, it is possible to obtain the result of the integration by direct examination without any calculations. The following are some simple methods of integration:


Integrating constant functions

When the integrand is a constant function , the integral is equal to the product of and the measure of the domain of integration. If and the domain is a subregion of , the integral gives the area of the region, while if the domain is a subregion of , the integral gives the volume of the region.
Example. Let and :D = \left\ in which case :\int_3^6 \int_2^4 \ 2 \ dx\, dy =2\int_3^6 \int_2^4 \ 1 \ dx\, dy= 2\cdot\operatorname(D) = 2 \cdot (2 \cdot 3) = 12, since by definition we have: :\int_3^6 \int_2^4 \ 1 \ dx\, dy=\operatorname(D).


Use of symmetry

When the domain of integration is symmetric about the origin with respect to at least one of the variables of integration and the integrand is
odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
with respect to this variable, the integral is equal to zero, as the integrals over the two halves of the domain have the same absolute value but opposite signs. When the integrand is
even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East ** Even language, a language spoken by the Evens * Odd and Even, a solitaire game w ...
with respect to this variable, the integral is equal to twice the integral over one half of the domain, as the integrals over the two halves of the domain are equal.
Example 1. Consider the function integrated over the domain :T=\left \, a disc with
radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
 1 centered at the origin with the boundary included. Using the linearity property, the integral can be decomposed into three pieces: :\iint_T \left(2\sin x - 3y^3 + 5\right) \, dx \, dy = \iint_T 2 \sin x \, dx \, dy - \iint_T 3y^3 \, dx \, dy + \iint_T 5 \, dx \, dy The function is an odd function in the variable and the disc is symmetric with respect to the -axis, so the value of the first integral is 0. Similarly, the function is an odd function of , and is symmetric with respect to the -axis, and so the only contribution to the final result is that of the third integral. Therefore the original integral is equal to the area of the disk times 5, or 5.
Example 2. Consider the function and as integration region the ball with radius 2 centered at the origin, :T = \left \. The "ball" is symmetric about all three axes, but it is sufficient to integrate with respect to -axis to show that the integral is 0, because the function is an odd function of that variable.


Normal domains on

This method is applicable to any domain for which: * the projection of onto either the -axis or the -axis is bounded by the two values, and * any line perpendicular to this axis that passes between these two values intersects the domain in an interval whose endpoints are given by the graphs of two functions, and . Such a domain will be here called a ''normal domain''. Elsewhere in the literature, normal domains are sometimes called type I or type II domains, depending on which axis the domain is fibred over. In all cases, the function to be integrated must be Riemann integrable on the domain, which is true (for instance) if the function is continuous.


-axis

If the domain is normal with respect to the -axis, and is a continuous function; then and (both of which are defined on the interval ) are the two functions that determine . Then, by Fubini's theorem: :\iint_D f(x,y)\, dx\, dy = \int_a^b dx \int_^ f(x,y)\, dy.


-axis

If is normal with respect to the -axis and is a continuous function; then and (both of which are defined on the interval ) are the two functions that determine . Again, by Fubini's theorem: :\iint_D f(x,y)\, dx\, dy = \int_a^b dy \int_^ f(x,y)\, dx.


Normal domains on

If is a domain that is normal with respect to the -plane and determined by the functions and , then :\iiint_T f(x,y,z) \, dx\, dy\, dz = \iint_D \int_^ f(x,y,z) \, dz\, dx\, dy This definition is the same for the other five normality cases on . It can be generalized in a straightforward way to domains in .


Change of variables

The limits of integration are often not easily interchangeable (without normality or with complex formulae to integrate). One makes a
change of variables Change or Changing may refer to: Alteration * Impermanence, a difference in a state of affairs at different points in time * Menopause, also referred to as "the change", the permanent cessation of the menstrual period * Metamorphosis, or change, ...
to rewrite the integral in a more "comfortable" region, which can be described in simpler formulae. To do so, the function must be adapted to the new coordinates.
Example 1a. The function is ; if one adopts the substitution , therefore , one obtains the new function .
* Similarly for the domain because it is delimited by the original variables that were transformed before ( and in example). * the differentials and transform via the absolute value of the determinant of the Jacobian matrix containing the partial derivatives of the transformations regarding the new variable (consider, as an example, the differential transformation in polar coordinates). There exist three main "kinds" of changes of variable (one in , two in ); however, more general substitutions can be made using the same principle.


Polar coordinates

In if the domain has a circular symmetry and the function has some particular characteristics one can apply the ''transformation to polar coordinates'' (see the example in the picture) which means that the generic points in Cartesian coordinates switch to their respective points in polar coordinates. That allows one to change the shape of the domain and simplify the operations. The fundamental relation to make the transformation is the following: :f(x,y) \rightarrow f(\rho \cos \varphi,\rho \sin \varphi ).
Example 2a. The function is and applying the transformation one obtains :f(x, y) = f(\rho \cos \varphi,\rho \sin \varphi) = \rho \cos \varphi + \rho \sin \varphi = \rho(\cos \varphi + \sin \varphi ).
Example 2b. The function is , in this case one has: :f(x, y) = \rho^2 \left(\cos^2 \varphi + \sin^2 \varphi\right) = \rho^2 using the Pythagorean trigonometric identity (very useful to simplify this operation).
The transformation of the domain is made by defining the radius' crown length and the amplitude of the described angle to define the intervals starting from .
Example 2c. The domain is , that is a circumference of radius 2; it's evident that the covered angle is the circle angle, so varies from 0 to 2, while the crown radius varies from 0 to 2 (the crown with the inside radius null is just a circle).
Example 2d. The domain is , that is the circular crown in the positive half-plane (please see the picture in the example); describes a plane angle while varies from 2 to 3. Therefore the transformed domain will be the following rectangle: :T = \. The
Jacobian determinant In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ...
of that transformation is the following: :\frac = \begin \cos \varphi & - \rho \sin \varphi \\ \sin \varphi & \rho \cos \varphi \end = \rho which has been obtained by inserting the partial derivatives of , in the first column respect to and in the second respect to , so the differentials in this transformation become . Once the function is transformed and the domain evaluated, it is possible to define the formula for the change of variables in polar coordinates: :\iint_D f(x,y) \, dx\, dy = \iint_T f(\rho \cos \varphi, \rho \sin \varphi) \rho \, d \rho\, d \varphi. is valid in the interval while , which is a measure of a length, can only have positive values.
Example 2e. The function is and the domain is the same as in Example 2d. From the previous analysis of we know the intervals of (from 2 to 3) and of (from 0 to ). Now we change the function: :f(x,y) = x \longrightarrow f(\rho,\varphi) = \rho \cos \varphi. finally let's apply the integration formula: :\iint_D x \, dx\, dy = \iint_T \rho \cos \varphi \rho \, d\rho\, d\varphi. Once the intervals are known, you have :\int_0^\pi \int_2^3 \rho^2 \cos \varphi \, d \rho \, d \varphi = \int_0^\pi \cos \varphi \ d \varphi \left \frac \right2^3 = \Big \sin \varphi \Big0^\pi \ \left(9 - \frac \right) = 0.


Cylindrical coordinates

In the integration on domains with a circular base can be made by the ''passage to
cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference d ...
''; the transformation of the function is made by the following relation: :f(x,y,z) \rightarrow f(\rho \cos \varphi, \rho \sin \varphi, z) The domain transformation can be graphically attained, because only the shape of the base varies, while the height follows the shape of the starting region.
Example 3a. The region is (that is the "tube" whose base is the circular crown of Example 2d and whose height is 5); if the transformation is applied, this region is obtained: :T = \ (that is, the parallelepiped whose base is similar to the rectangle in Example 2d and whose height is 5). Because the component is unvaried during the transformation, the differentials vary as in the passage to polar coordinates: therefore, they become . Finally, it is possible to apply the final formula to cylindrical coordinates: :\iiint_D f(x,y,z) \, dx\, dy\, dz = \iiint_T f(\rho \cos \varphi, \rho \sin \varphi, z) \rho \, d\rho\, d\varphi\, dz. This method is convenient in case of cylindrical or conical domains or in regions where it is easy to individuate the ''z'' interval and even transform the circular base and the function.
Example 3b. The function is and as integration domain this
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...
: . The transformation of in cylindrical coordinates is the following: :T = \. while the function becomes :f(\rho \cos \varphi, \rho \sin \varphi, z) = \rho^2 + z Finally one can apply the integration formula: :\iiint_D \left(x^2 + y^2 +z\right) \, dx\, dy\, dz = \iiint_T \left( \rho^2 + z\right) \rho \, d\rho\, d\varphi\, dz; developing the formula you have :\int_^5 dz \int_0^ d\varphi \int_0^3 \left( \rho^3 + \rho z \right)\, d\rho = 2 \pi \int_^5 \left \frac + \frac \right0^3 \, dz = 2 \pi \int_^5 \left( \frac + \frac z\right)\, dz = \cdots = 405 \pi.


Spherical coordinates

In some domains have a spherical symmetry, so it's possible to specify the coordinates of every point of the integration region by two angles and one distance. It's possible to use therefore the ''passage to
spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
''; the function is transformed by this relation: :f(x,y,z) \longrightarrow f(\rho \cos \theta \sin \varphi, \rho \sin \theta \sin \varphi, \rho \cos \varphi) Points on the -axis do not have a precise characterization in spherical coordinates, so can vary between 0 and 2. The better integration domain for this passage is the sphere.
Example 4a. The domain is (sphere with radius 4 and center at the origin); applying the transformation you get the region :T = \. The Jacobian determinant of this transformation is the following: :\frac = \begin \cos \theta \sin \varphi & - \rho \sin \theta \sin \varphi & \rho \cos \theta \cos \varphi \\ \sin \theta \sin \varphi & \rho \cos \theta \sin \varphi & \rho \sin \theta \cos \varphi \\ \cos \varphi & 0 & - \rho \sin \varphi \end = \rho^2 \sin \varphi The differentials therefore are transformed to . This yields the final integration formula: :\iiint_D f(x,y,z) \, dx\, dy\, dz = \iiint_T f(\rho \sin \varphi \cos \theta, \rho \sin \varphi \sin \theta, \rho \cos \varphi) \rho^2 \sin \varphi \, d\rho\, d\theta\, d\varphi.
It is better to use this method in case of spherical domains and in case of functions that can be easily simplified by the first fundamental relation of trigonometry extended to (see Example 4b); in other cases it can be better to use cylindrical coordinates (see Example 4c). :\iiint_T f(a,b,c) \rho^2 \sin \varphi \, d\rho\, d\theta\, d\varphi. The extra and come from the Jacobian. In the following examples the roles of and have been reversed.
Example 4b. is the same region as in Example 4a and is the function to integrate. Its transformation is very easy: :f(\rho \sin \varphi \cos \theta, \rho \sin \varphi \sin \theta, \rho \cos \varphi) = \rho^2, while we know the intervals of the transformed region from : :T=\. We therefore apply the integration formula: :\iiint_D \left(x^2 + y^2 + z^2\right) \, dx\, dy\, dz = \iiint_T \rho^2 \, \rho^2 \sin \theta \, d\rho\, d\theta\, d\varphi, and, developing, we get :\iiint_T \rho^4 \sin \theta \, d\rho\, d\theta\, d\varphi = \int_0^ \sin \varphi \,d\varphi \int_0^4 \rho^4 d \rho \int_0^ d\theta = 2 \pi \int_0^ \sin \varphi \left \frac \right0^4 \, d \varphi = 2 \pi \left \frac \right0^4 \Big \cos \varphi \Big0^ = \frac.
Example 4c. The domain is the ball with center at the origin and radius , :D = \left \ and is the function to integrate. Looking at the domain, it seems convenient to adopt the passage to spherical coordinates, in fact, the intervals of the variables that delimit the new region are obviously: :T=\. However, applying the transformation, we get :f(x,y,z) = x^2 + y^2 \longrightarrow \rho^2 \sin^2 \theta \cos^2 \varphi + \rho^2 \sin^2 \theta \sin^2 \varphi = \rho^2 \sin^2 \theta. Applying the formula for integration we obtain: :\iiint_T \rho^2 \sin^2 \theta \rho^2 \sin \theta \, d\rho\, d\theta\, d\varphi = \iiint_T \rho^4 \sin^3 \theta \, d\rho\, d\theta\, d\varphi which can be solved by turning it into an iterated integral. \iiint_T \rho^4 \sin^3 \theta \, d\rho\, d\theta\, d\varphi = \underbrace_ \,\underbrace_\, \underbrace_. I = \left.\int_0^\rho^4 d\rho = \frac\right\vert_0^ = \fraca^5, II = \int_0^\pi \sin^3\theta \, d\theta = -\int_0^\pi \sin^2\theta \, d(\cos \theta) = \int_0^\pi (\cos^2\theta-1) \, d(\cos \theta) = \left.\frac\^\pi_0 - \left.\cos\theta\^\pi_0 = \frac, III = \int_0^ d \varphi = 2\pi. Collecting all parts, \iiint_T \rho^4 \sin^3 \theta \, d\rho\, d\theta\, d\varphi = I\cdot II\cdot III = \fraca^5\cdot \frac\cdot 2\pi = \frac\pi a^5. Alternatively, this problem can be solved by using the passage to cylindrical coordinates. The new intervals are :T=\left\; the interval has been obtained by dividing the ball into two
hemisphere Hemisphere refers to: * A half of a sphere As half of the Earth * A hemisphere of Earth ** Northern Hemisphere ** Southern Hemisphere ** Eastern Hemisphere ** Western Hemisphere ** Land and water hemispheres * A half of the (geocentric) celes ...
s simply by solving the
inequality Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...
from the formula of (and then directly transforming into ). The new function is simply . Applying the integration formula :\iiint_T \rho^2 \rho \, d \rho \, d \varphi \, dz. Then we get :\begin \int_0^ d\varphi \int_0^ \rho^3 d\rho \int_^\, dz &= 2 \pi \int_0^ 2 \rho^3 \sqrt \, d\rho \\ &= -2 \pi \int_^0 (9 a^2 - t) \sqrt\, dt && t = 9 a^2 - \rho^2 \\ &= 2 \pi \int_0^ \left ( 9 a^2 \sqrt - t \sqrt \right ) \, dt \\ &= 2 \pi \left( \int_0^ 9 a^2 \sqrt \, dt - \int_0^ t \sqrt \, dt\right) \\ &= 2 \pi \left a^2 \frac23 t^ - \frac t^ \right0^ \\ &= 2 \cdot 27 \pi a^5 \left ( 6 - \frac \right ) \\ &= \frac a^5. \end Thanks to the passage to cylindrical coordinates it was possible to reduce the triple integral to an easier one-variable integral.
See also the differential volume entry in nabla in cylindrical and spherical coordinates.


Examples


Double integral over a rectangle

Let us assume that we wish to integrate a multivariable function over a region : :A = \left \ \mbox f(x,y) = x^2 + 4y\, From this we formulate the iterated integral :\int_7^ \int_^ (x^2 + 4y) \, dx\, dy The inner integral is performed first, integrating with respect to and taking as a constant, as it is not the variable of integration. The result of this integral, which is a function depending only on , is then integrated with respect to . :\begin \int_^ \left(x^2 + 4y\right) \, dx & = \left frac13 x^3 + 4yx \right^ \\ &= \frac13(14)^3 + 4y(14) - \frac13(11)^3 - 4y(11) \\ &= 471 + 12y \end We then integrate the result with respect to . :\begin \int_7^ (471 + 12y) \ dy & = \Big 71y + 6y^2\Big^ \\ &= 471(10)+ 6(10)^2 - 471(7) - 6(7)^2 \\ &= 1719 \end In cases where the double integral of the absolute value of the function is finite, the order of integration is interchangeable, that is, integrating with respect to ''x'' first and integrating with respect to ''y'' first produce the same result. That is
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 ...
. For example, doing the previous calculation with order reversed gives the same result: : \begin \int_^ \int_^ \, \left(x^2 + 4y\right) \, dy\, dx & = \int_^ \Big ^2 y + 2y^2 \Big^ \, dx \\ &= \int_^ \, (3x^2 + 102) \, dx \\ &= \Big ^3 + 102x \Big^ \\ &= 1719. \end


Double integral over a normal domain

Consider the region (please see the graphic in the example): :D = \ Calculate :\iint_D (x+y) \, dx \, dy. This domain is normal with respect to both the ''x''- and ''y''-axes. To apply the formulae it is required to find the functions that determine ''D'' and the intervals over which these functions are defined. In this case the two functions are: :\alpha (x) = x^2\text\beta (x) = 1 while the interval is given by the intersections of the functions with ''x'' = 0, so the interval is 'a'', ''b''= , 1(normality has been chosen with respect to the ''x''-axis for a better visual understanding). It is now possible to apply the formula: :\iint_D (x+y) \, dx \, dy = \int_0^1 dx \int_^1 (x+y) \, dy = \int_0^1 dx \ \left y + \frac \right1_ (at first the second integral is calculated considering ''x'' as a constant). The remaining operations consist of applying the basic techniques of integration: :\int_0^1 \left y + \frac\right1_ \, dx = \int_0^1 \left(x + \frac - x^3 - \frac \right) dx = \cdots = \frac. If we choose normality with respect to the ''y''-axis we could calculate :\int_0^1 dy \int_0^ (x+y) \, dx. and obtain the same value.


Calculating volume

Using the methods previously described, it is possible to calculate the volumes of some common solids. *
Cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...
: The volume of a cylinder with height and circular base of radius can be calculated by integrating the constant function over the circular base, using polar coordinates. ::\mathrm = \int_0^ d \varphi\, \int_0^R h \rho \, d \rho = 2 \pi h \left frac\right0^R = \pi R^2 h This is in agreement with the formula for the volume of a
prism Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimentary ...
::\mathrm = \text \times \text. *
Sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
: The volume of a sphere with radius can be calculated by integrating the constant function 1 over the sphere, using spherical coordinates. ::\begin \text &= \iiint_D f(x,y,z) \, dx\, dy\, dz \\ &= \iiint_D 1 \, dV \\ &= \iiint_S \rho^2 \sin \varphi \, d\rho\, d\theta\, d\varphi \\ &= \int_0^ \, d \theta \int_0^ \sin \varphi\, d \varphi \int_0^R \rho^2\, d \rho \\ &= 2 \pi \int_0^\pi \sin \varphi\, d \varphi \int_0^R \rho^2\, d \rho \\ &= 2 \pi \int_0^\pi \sin \varphi \frac\, d \varphi \\ &= \frac23 \pi R^3 \Big \cos \varphi\Big0^\pi = \frac43 \pi R^3. \end *
Tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...
(triangular
pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...
or 3- simplex): The volume of a tetrahedron with its apex at the origin and edges of length along the -, - and -axes can be calculated by integrating the constant function 1 over the tetrahedron. ::\begin \text &= \int_0^\ell dx \int_0^\, dy \int_0^\, dz \\ &= \int_0^\ell dx \int_0^ (\ell - x - y)\, dy \\ &= \int_0^\ell \left( l^2 - 2 \ell x + x^2 - \frac\right)\, dx \\ &= \ell^3 - \ell \ell^2 + \frac - \left frac - \frac + \frac\right0^ \ell \\ &= \frac - \frac = \frac\end :This is in agreement with the formula for the volume of a
pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...
::\mathrm = \frac13 \times \text \times \text = \frac13 \times \frac \times \ell = \frac.


Multiple improper integral

In case of unbounded domains or functions not bounded near the boundary of the domain, we have to introduce the double
improper integral In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some instances as both endpoin ...
or the triple improper integral.


Multiple integrals and iterated integrals

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 if :\iint_ \left, f(x,y)\\,d(x,y)<\infty, that is, if the integral is absolutely convergent, then the multiple integral will give the same result as either of the two iterated integrals: :\iint_ f(x,y)\,d(x,y)=\int_A\left(\int_B f(x,y)\,dy\right)\,dx=\int_B\left(\int_A f(x,y)\,dx\right)\,dy. In particular this will occur if is a bounded function and and are bounded sets. If the integral is not absolutely convergent, care is needed not to confuse the concepts of ''multiple integral'' and ''iterated integral'', especially since the same notation is often used for either concept. The notation :\int_0^1\int_0^1 f(x,y)\,dy\,dx means, in some cases, an iterated integral rather than a true double integral. In an iterated integral, the outer integral :\int_0^1 \cdots \, dx is the integral with respect to of the following function of : :g(x)=\int_0^1 f(x,y)\,dy. A double integral, on the other hand, is defined with respect to area in the -plane. If the double integral exists, then it is equal to each of the two iterated integrals (either "" or "") and one often computes it by computing either of the iterated integrals. But sometimes the two iterated integrals exist when the double integral does not, and in some such cases the two iterated integrals are different numbers, i.e., one has :\int_0^1\int_0^1 f(x,y)\,dy\,dx \neq \int_0^1\int_0^1 f(x,y)\,dx\,dy. This is an instance of rearrangement of a
conditionally convergent In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely. Definition More precisely, a series of real numbers \sum_^\infty a_n is said to converge conditionally if \lim_\,\s ...
integral. On the other hand, some conditions ensure that the two iterated integrals are equal even though the double integral need not exist. By the Fichtenholz– Lichtenstein theorem, if is bounded on and both iterated integrals exist, then they are equal. Moreover, existence of the inner integrals ensures existence of the outer integrals. The double integral need not exist in this case even as
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 ...
, according to Sierpiński. The notation :\int_ f(x,y)\,dx\,dy may be used if one wishes to be emphatic about intending a double integral rather than an iterated integral.


Triple integral

Main article:
Volume integral In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many ...
Triple integral was demonstrated by Fubini's theorem. Drichlet theorem and Liouville 's extension theorem on Triple integral.


Some practical applications

Quite generally, just as in one variable, one can use the multiple integral to find the average of a function over a given set. Given a set and an integrable function over , the average value of over its domain is given by :\bar = \frac \int_D f(x)\, dx, where is the measure of . Additionally, multiple integrals are used in many applications in
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 ...
. The examples below also show some variations in the notation. In
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to object ...
, the moment of inertia is calculated as the volume integral (triple integral) of the
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
weighed with the square of the distance from the axis: :I_z = \iiint_V \rho r^2\, dV. The
gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...
associated with a mass distribution given by a mass measure on three-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
is :V(\mathbf) = -\iiint_ \frac\,dm(\mathbf). If there is a continuous function representing the density of the distribution at , so that , where is the Euclidean
volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form :dV ...
, then the gravitational potential is :V(\mathbf) = -\iiint_ \frac\,\rho(\mathbf)\,d^3\mathbf. In
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 ...
,
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
can be written using multiple integrals to calculate the total magnetic and electric fields. In the following example, the electric field produced by a distribution of charges given by the volume
charge density In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in ...
is obtained by a ''triple integral'' of a vector function: :\vec E = \frac \iiint \frac \rho (\vec r')\, d^3 r'. This can also be written as an integral with respect to a
signed measure In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values. Definition There are two slightly different concepts of a signed measure, depending on whether or not ...
representing the charge distribution.


See also

* Main
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
theorems that relate multiple integrals: ** Divergence theorem ** Stokes' theorem ** Green's theorem


References


Further reading

* * *Herman, Edwin “Jed” & Strang, Gilbert (2016): ''Calculus : Volume 3'' : OpenStax, Rice University, Houston, Texas, USA. .
PDF


External links

* *
Mathematical Assistant on Web
online evaluation of double integrals in Cartesian coordinates and
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
(includes intermediate steps in the solution, powered by
Maxima (software) Maxima () is a computer algebra system (CAS) based on a 1982 version of Macsyma. It is written in Common Lisp and runs on all POSIX platforms such as macOS, Unix, BSD, and Linux, as well as under Microsoft Windows and Android. It is free softwa ...
) {{Authority control Integral calculus Multivariable calculus de:Integralrechnung#Mehrdimensionale Integration