Multiple Integral - Mathematical Definition

Mathematical Definition

For n > 1, consider a so-called "half-open" n-dimensional hyperrectangular domain T, defined as:

Partition each interval [a_j, b_j) into a finite family I_j of non-overlapping subintervals i_jα, with each subinterval closed at the left end, and open at the right end.

Then the finite family of subrectangles C given by

is a partition of T; that is, the subrectangles C_k are non-overlapping and their union is T.

Let f : T → R be a function defined on T. Consider a partition C of T as defined above, such that C is a family of m subrectangles C_m and

We can approximate the total nth-dimensional volume bounded below by T and above by f with the following Riemann sum:

where P_k is a point in C_k and m(C_k) is the product of the lengths of the intervals whose Cartesian product is C_k, otherwise known as the measure of C_k.

The diameter of a subrectangle C_k is the largest of the lengths of the intervals whose Cartesian product is C_k. The diameter of a given partition of T is defined as the largest of the diameters of the subrectangles in the partition. Intuitively, as the diameter of the partition C is restricted smaller and smaller, the number of subrectangles m gets larger, and the measure m(C_k) of each subrectangle grows smaller. The function f is said to be Riemann integrable if the limit

exists, where the limit is taken over all possible partitions of T of diameter at most δ.

If f is Riemann integrable, S is called the Riemann integral of f over T and is denoted

Frequently this notation is abbreviated as

where x represents the n-tuple (x₁, ... x_n) and dx is the n-dimensional volume differential.

The Riemann integral of a function defined over an arbitrary bounded n-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 n dimensions will be called multiple integral.