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Mathematical definition

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

Partition each interval [ a, b) into a finite family I of non-overlapping subintervals i, 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 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 and

We can approximate the total (n + 1) -dimensional volume bounded below by the n -dimensional hyperrectangle T and above by the n -dimensional graph of f with the following Riemann sum:

where P is a point in C and m(C) is the product of the lengths of the intervals whose Cartesian product is C, also known as the measure of C.

The diameter of a subrectangle C is the largest of the lengths of the intervals whose Cartesian product is C. 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) 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 (x,..., x) and d x 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 the multiple integral.