In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in (the real-number plane) are called double integrals, and integrals of a function of three variables over a region in (real-number 3D space) are called triple integrals. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.
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 (on the three-dimensional Cartesian plane where ) 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 (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:
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 (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.
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.