Solve the following integral
$$\int_0^\infty \int_0^\infty (x^2 + y^2)e^{-(x^2+y^2)} \ dxdy $$
For me, it is clear that we can use polar coordinates to solve this integral quickly (and yes, we can do it without it). The problem is, I don't know how to set up the region for the new integral in polar coordinates. Precisely, I don't understand how to set up $\theta$. If someone could explain to me, maybe visually, I would appreciate that.
$$\int_?^? \int_0^\infty r^3e^{-r^2} \ drd\theta $$
*Self-studying
To start with, realize what is the original region where we are integrating.
Let us denote it by $R$. Then we have:
\begin{align*} R = \{(x,y)\in\mathbb{R}^{2} : (x\geq 0)\wedge(y\geq 0)\} \end{align*} which coincides exactly with the first quadrant.
Hence the length $r$ ranges from $0$ to $+\infty$ and the angle ranges from $0$ to $\pi/2$.
More precisely, the proposed integral can be rewritten as \begin{align*} \iint_{R}(x^{2} + y^{2})e^{-x^{2}-y^{2}}\mathrm{d}x\mathrm{d}y = \int_{0}^{\pi/2}\mathrm{d}\theta\int_{0}^{\infty}r^{3}e^{-r^{2}}\mathrm{d}r \end{align*}
Hopefully this helps !