How to evaluate bounds by converting to polar coordinates?

Question

My question is only in regards to bounds of the following: I am trying to switch between polar coordinates and so using x=rcos(0) and y=rsin(0); however, the large source of my confusion lies in what the new bounds becomes. This integral will eventually use a u-substitution (and I'll get something like) then I can evaluate it before taking it's second integral, but how do I evaluate the changing bounds?

Answer 1

I tell my students to draw a picture of the region. Don't try to find the limits algebraically. The region in the original integral is a quarter circle of radius 3, which is pretty easy to set up in polar.