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# Multiple Integral

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The integral of a continuous function on a regular region can be defined as limit of Riemann-sums:

Here is approximated by the union of elementary volumes (usually simplices or parallelepipeds) such that

denotes the maximal diameter of the sets , and are arbitrary points in . The sets are disjoint except possibly at portions of their boundaries.

The notation symbolizes the approximation process, and is called volume element. A shorter notation is , or, more detailed,

to point out the variables of integration.

For multiple integrals are called double integrals, for triple integrals. For double integrals one also uses the notation

or

and, for triple integrals,

or

Because of the continuity of the integrand , the definition of the multiple integral is independent of the choice of the elementary volumes as well as of the points .

For a positive function, the integral coincides with the volume of the set

In particular, is just the volume of the region of integration .

In order to garantuee the existence of multiple integrals, weaker conditions on continuity and smoothness of and are possible. The integral can also exist if the region of integration is unbounded. Such an integral is called an improper integral.