Originally Posted by

**mathisfunforme** This is not a homework problem; I'm just making sure I can follow all the logic to this problem.

Hi I have the following part of my book which I am trying to understand but I am stuck on a single part of the conversion from rectangular coordinates to polar coordinates:

The book lists:

$\displaystyle (\Gamma(1/2))^2 = 4 \int_0^{\frac{\pi}{2}} \int_0^\infty e^{-r^2}r dr d\theta = 4\frac{\pi}{2} (\frac{e^{-r^2}}{-2}) \ evaluated \ from \ 0 \ to \ \infty $

When I do a u substitution for the improper integral how do I get evaluate the integral with lim as $\displaystyle t \rightarrow \infty \ for \int_0^t e^u du $ I have set $\displaystyle u = -r^2, du = -2rdr $.

The result is $\displaystyle \pi $

Thank you.