How to integrate delta function?

Question

There is this question that ask me to compute

by performing a change of coordinates to polar coordinates.

My attempts:

I manage to change it to $$I=\int_0^\infty \int_0^{2\pi} re^{-r}δ(r-R)d\theta\,dr$$ but the problem is how do I integrate delta function?

Answer 1

$$I~=~\int_{[0,2\pi]}\! d\theta \int_{\mathbb{R}_+} \! dr~ re^{-r}~\delta (r-R) ~=~2\pi \int_{\mathbb{R}} \! dr~H(r) re^{-r}~\delta (r-R)~=~2\pi H(R)Re^{-R}, $$ where $H$ denotes the Heaviside step function.