Multiple integral


In mathematics, a multiple integral is a definite integral of a function of several real variables, for instance, or.
Integrals of a function of two variables over a region in are called double integrals, and integrals of a function of three variables over a region in are called triple integrals.

Introduction

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 -axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function and the plane which contains its domain. If there are more variables, a multiple integral will yield hypervolumes 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, followed by the function and integrand arguments in proper order. 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:
Since the concept of an antiderivative is only defined for functions of a single real variable, the usual definition of the indefinite integral does not immediately extend to the multiple integral.

Mathematical definition

For, consider a so-called "half-open" -dimensional hyperrectangular domain, defined as
Partition 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
is a partition 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
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:
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
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
Frequently this notation is abbreviated as
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. 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.

Particular cases

In the case of the integral
is the double integral of on, and if the integral
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, a series of integrals of one variable, each being directly solvable. For continuous functions, this is justified by Fubini's theorem. 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
in which case
since by definition we have:

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 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 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
a disc with radius 1 centered at the origin with the boundary included.
Using the linearity property, the integral can be decomposed into three pieces:
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,
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 if the function is continuous.

-axis

If the domain is normal with respect to the -axis, and is a continuous function; then and are the two functions that determine. Then, by Fubini's theorem:

-axis

If is normal with respect to the -axis and is a continuous function; then and are the two functions that determine. Again, by Fubini's theorem:

Normal domains on

If is a domain that is normal with respect to the -plane and determined by the functions and, then
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. One makes a change of variables 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.

There exist three main "kinds" of changes of variable ; 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 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:
Example 2a. The function is and applying the transformation one obtains
Example 2b. The function is, in this case one has:
using the Pythagorean trigonometric identity.

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.

Example 2d. The domain is, that is the circular crown in the positive half-plane ; describes a plane angle while varies from 2 to 3. Therefore the transformed domain will be the following rectangle:
The Jacobian determinant of that transformation is the following:
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:
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 and of . Now we change the function:
Finally let's apply the integration formula:
Once the intervals are known, you have