# Math Help - improper integral problem...

1. ## improper integral problem...

how to evaluate the improper integral using polar coordinates?

2. Hello,
Originally Posted by iwonder
how to evaluate the improper integral using polar coordinates?
$x=r \cos \theta$
$y=r \sin \theta$

Then $x^2+y^2=r^2 (\cos^2 \theta+ \sin^2 \theta)=r^2$

and the Jacobian of the transformation is r.
So $dx ~ dy=r ~ dr ~ d \theta$

Since $(x,y) \in \mathbb{R} \times \mathbb{R}$, and since r is positive, we have $r \in [0,+ \infty)$ and $\theta \in [0,2 \pi]$, so that it covers all the plane.

---> $=\int_0^{2 \pi} \int_0^\infty e^{-3r^2} ~ r ~ dr ~ d \theta$

$=\left(\int_0^{2 \pi} ~ d \theta \right) \cdot \left(\int_0^\infty e^{-3r^2} ~ r ~ dr\right)$

In order to calculate the second one, make a substitution : $u=r^2$