One way to prove that the area under the graph of the Gaussian function $e^{-x^2}$ is $\sqrt{\pi}$ is to compute the quantity $\int_{-\infty}^\infty \int_{-\infty}^\infty e^{-x^2-y^2}$ using a change to polar coordinates. However, this proof uses the fact that
$$\int_{-\infty}^\infty \int_{-\infty}^\infty f(x,y) \thinspace dy \thinspace dx = \int_0^{2\pi} \int_0^\infty f(r\cos\theta, r\sin\theta)r \thinspace dr \thinspace d\theta.$$
The only calculus text I own is Stewart's Calculus, which does not go into improper integrals in polar coordinates. Where can I find a proper statement and proof of this fact?