This integral is bounded by

and

Also by

and

If you graph the domain you will see we are bounded by the line y=x and the roof h. Of course, the line y=x is associated with

and this would then be our start theta value. We are then bounded by x=0 or the y-axis.

We are not bounded by a cylinder, so we must find an equation of r that relates to theta. This can be done by looking at your domain and realising that

We then have

Of course

Why in gods name would you want to do this question in polar I have no idea. But this should be good