When should I use a double integral?

Question

Under what conditions should I use a double integral to find the area under a graph and not simple integration?

Answer 1

In lots of problems, evaluating the inside integral in a double integral just gives you what you would have written down for the single integral. But it's at least of theoretical interest that the area equals $\iint 1 \; dA.$

However, in vector calculus, there are great tricks for finding areas that start with writing area $=\iint 1\; dA,$ and then applying Green's theorem or converting to polar coordinates. So my short answer is that "often, the double integral is a good place to start."