# Change the integral into polar coordinates

Convert the integral $\int_{-2}^{2} \int_0^{\sqrt{4-x^2}} e^{-(x^2+y^2)} dy\,dx$ into polar coordinates don't need to solve.

my solution is currently $\int_0^2 \int_0^\pi re^{-r^2} d\theta\, dr$

not sure if this is correct, I have subbed in $r\cos\theta$ and $r\sin\theta$ and multiplied by the jacobian, not sure if limits are correct, keep getting confused as to which way around they go, thanks in advance.

Yes it is absolutely correct indeed we are integrating $e^{-(x^2+y^2)}$ on half circle with radius $2$ centered at the origin with $y\ge 0$ and therefore
• $\theta \in [0,\pi]$
• $r \in [0,2]$
and you have correctly transformed $e^{-(x^2+y^2)}$ in $e^{-r^2}$ taking into account the $r$ factor from the Jacobian.