Integration limits of the double integral after conversion to the polar coordinates

Question

I want to solve the following double integral:

$$\int_0^{\infty}dx\int_{-\infty}^{\infty}dy\,f(x,y).$$

And for example I made a conversion to the polar coordinates, $x=r\cos{\theta}$ and $y=r\sin{\theta}$ and get a new integral where I plugged in new variables and multiplied the Integral with the Jacobian factor, in this case $r$. How could I find new integration limits in terms of $r$ and $\theta$ based on the integration limits in terms of $x$ and $y$?

Answer 1

$$r:0 \to \infty \\\theta : - \pi/2 \to \pi/2$$