I ran into some trouble with some integrals that use polar coordinates.

Sounds fair!

....

The region for integration is a triangle which I thought was a little weird. I think the r limits start from 0 to something (I called it b in the integral below), but I think the

limits are

.

So, without the final r limit, I have

.

My problem with this one is that the region of integration is not a circle centred at the origin but rather the point

. I am aware that the region is a semi circle. For this reason, are the

limits

?

Are the r limits

?

However, the integrand is going to be

.

So overall, I get that the integral should be

.

Clearly the r limits are incorrect!

From here, I decided to put

into the upper limit of the double integral:

.

Is this integral the correct one to evaluate?