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

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 $y$?

$$r:0 \to \infty \\\theta : - \pi/2 \to \pi/2$$
• Your region is the right half plane. You have to cover it using the new limits.For $r$, think of a line starting from the origin and enters your region and leave it. For $\theta$, this line should rotate to cover the region, the angle $\theta$ measured with positive x-axis in positive direction Feb 3 '14 at 13:20