Chapter 13.3, Problem 65E

Improper integrals Improper integrals arise in polar coordinates when the radial coordinate r becomes arbitrarily large. Under certain conditions, these integrals are treated in the usual way:

     ${\displaystyle {\int }_{\alpha }^{\beta }{\displaystyle {\int }_a^{\infty }f\left(r,\theta \right)}}rdrd\theta =\underset{b\to \infty }{\lim }{\displaystyle {\int }_{\alpha }^{\beta }{\displaystyle {\int }_a^{b}f\left(r,\theta \right)}}rdrd\theta .$

Use this technique to evaluate the following integrals.

65.     ${\displaystyle \underset{R}{\iint }{e}^{-x^2-y^2}dA;R=\left\{\left(r,\theta \right):0\le r<\infty ,0\le \theta \le \pi /2\right\}}$