I need to solve the integral in polar coordinates $$\int_0^1\int_0^\sqrt{1-x^2}e^{-x^2-y^2}dydx$$
What I've tried:
$$x= r\cos(\theta), y = r\sin(\theta) \Rightarrow e^{-r^2} $$ $$|J| = r$$
Therefore,
$$\int_{a}^{b} \int_{c}^{d} e^{-r^2}rdrd\theta$$
But I don't know what to do with the lower and upper limits of the integral $a, b, c, d$. I've tried doing the following: $0 = r\sin(\theta), r = c = 0$ and $y = \sqrt{1-x^2} \iff r\sin(\theta) = \sqrt{1-r^2\cos^2(\theta)} \iff r = 1$ so $d$ would be $d = 1$.
But I'm not sure I'm setting these bounds correctly. How would it go for the limits of $\theta$?