The first task in evaluating a multiple integral is to reduce it to an iterated integral, which can be performed by sequential ordinary integrations. Very occasionally one can go the other way: taking an iterated integral and, by writing it as a multiple integral, and evaluating it using different coordinates, one can convert a mysterious integral into a formula.
The classical example is our second example above. If we square a "Gaussian integral", consider the result an integral over the x - y plane, and reduce that to a multiple integral in polar coordinates, we can evaluate the thing, and deduce a formula for the Gaussian Integral. This derivation appears on T-shirts available from the department.

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