Solving Iterated Integrals

While the order of the differentials, dx dy or dy dx, does not matter in a double integral, it is important in an iterated integral. The inside integral sign goes with the inside differential, and is performed first.

When the region D is a rectangle, there are two possible orders of integration, because all the boundaries are constant. Thus there are two different iterated integrals over a rectangle. Integrating first with respect to y we have

and integrating first with respect to x we have

Using the Iterated Integral Theorem twice, we see that both iterated integrals must equal the double integral.

Therefore the two iterated integrals are equal to each other. We have proved a corollary.

Discussion This corollary is the simplest form of a result known as Fubini's Theorem. Remember that by our Permanent Assumption, f(x, y) is continuous on D, For an idea of the difficulties that arise when f(x, y) is not assumed to be continuous, see Problem 49 at the end of this section.

There are also other regions besides rectangles over which we can integrate in either of two orders, such as Example 5 in this section.

In the following two examples we evaluate the double integrals which were approximated by double Riemann sums in the preceding section.

Example 1

Example 2

Example 3

Example 4