Evaluating $\int_0^1 \int_0^{\sqrt{1-x^2}}e^{-(x^2+y^2)} \, dy \, dx\ $ using polar coordinates Use polar coordinates to evaluate $\int_0^1 \int_0^{\sqrt{1-x^2}}e^{-(x^2+y^2)} \, dy \, dx\
$
I understand that we need to change $x^2+y^2$ to $r^2$ and then we get $\int_0^1 \int_0^{\sqrt{1-x^2}} e^{-(r^2)} \, dy \, dx\
$. Then I know I need to change the bounds with respect to $dy$ but I am unsure on how to do that and further. Please help me.

Hints:
$$\bullet\;\;\;x=r\cos\theta\;,\;\;y=r\sin\theta\;,\;0\le\theta\le \frac\pi2\;\text{(why?). The Jacobian is}\;\;|J|=r$$
So the integral is
$$\int\limits_0^1\int\limits_0^{\pi/2}re^{-r^2}drd\theta$$
