changing bounds from rectangular to polar

i just have a general question about how to change the bounds of integration from rectangular to polar. is it as simple as just substituting in y=rsinx (x will represent theta since i don't know how to make theta) and x=rcosx? or do i need to know what the bounds look like and then remake them based on that?

for instance, if a draw a picture of the rectangular bounds, would i then use that picture to help me figure out what the new polar bounds are?

so let's say i have an integral bounded by y <x< root(2-y^2)

would i simply plug in the polar formulas for y, or would i draw out those bounds, look at the picture, and figure out what the proper polar coordinates would be? i'm having trouble doing it either way, as i can't figure out what the polar coordinates would be even after drawing out the picture. and i don't know how to figure out the exact angles that theta is bounded by. if you have any advice, i would greatly appreciate it. i can do most of the polar integration except for these instances when i have to change the bounds to polar.