Multiple Integral - Multiple Integrals and Iterated Integrals

Multiple Integrals and Iterated Integrals

See also: Order of integration (calculus)

Fubini's theorem states that if

that is, if the integral is absolutely convergent, then the multiple integral will give the same result as the iterated integral,

In particular this will occur if |f(x,y)| is a bounded function and A and B are bounded sets.

If the integral is not absolutely convergent, care is needed not to confuse the concepts of multiple integral and iterated integral, especially since the same notation is often used for either concept. The notation

means, in some cases, an iterated integral rather than a true double integral. In an iterated integral, the outer integral

is the integral with respect to x of the following function of x:

A double integral, on the other hand, is defined with respect to area in the xy-plane. If the double integral exists, then it is equal to each of the two iterated integrals (either "dy dx" or "dx dy") and one often computes it by computing either of the iterated integrals. But sometimes the two iterated integrals exist when the double integral does not, and in some such cases the two iterated integrals are different numbers, i.e., one has

This is an instance of rearrangement of a conditionally convergent integral.

The notation

may be used if one wishes to be emphatic about intending a double integral rather than an iterated integral.