We can integrate over regions more general than rectangles by chopping up into sets of small diameter and approximating by rectangles. It is also possible to allow to be an unbounded region, provided that satisfies suitable conditions.
To show that
Solution. We note that the inner integral can be immediately described using the Laplace transform of (or rather, ): it is . (Alternatively, we can integrate by parts as in 1.47-8.)
Thus the double integral is