You can prove that integral(-inf-->+inf) exp(-x^2) dx = sqrt(pi) by converting (squaring it) to a double integral or you can use the residue theorem after converting it to a complex integral. The first way is easier.
Yup, but Complex variable integration can integrate lots of integrals you never thought was possible at the beginning; the residue theorem is just great. And Cauchy's theorems about integrating f over a region/closed contour... where f is holomorfic, are quite funny if you're used to real...