Converting bounds of double integral to polar coordinates I'm trying to convert the following to polar coordinates:
$$\int_0^\infty \int_{-\infty}^{-x}\frac{1}{2\pi}e^{-(x^2+y^2)/2}\,dx\,dy$$
After converting to polar coordinates, it should be:
$$\int_0^\infty \int_{(3/2)\pi}^{(7/4)\pi}\frac{1}{2\pi}e^{-r^2/2}\,dr\,d\theta$$
My question is how do we arrive at the bounds of $(3/2)\pi$ and $(7/4)\pi$. 

The first integral should be
$$\large\int_{0}^{\infty}dx\int_{-\infty}^{-x}\frac{1}{2\pi}e^{-(x^2+y^2)/2}dy$$
wich represent the integral over the half of the 4 quadrant between $y$ axis and $y=-x$ that is for $\theta$ between $3\pi/2$ and $7\pi/4$.
Note also that here
$$\large\int_{0}^{\infty}dr\int_{(3/2)\pi}^{(7/4)\pi}\frac{1}{2\pi}\color{red}re^{-r^2/2}d\theta$$
we need an "$r$" extra term for polar coordinates jacobian wich makes this kind of integral easy to be solved by polar coordinates.
