I don't understand how the limits of integration have been obtained in the following worked example:

So, how did they get the limits ?

I know that for the double integral they've used polar coordinates. The represents the unit circle. I tried to sketch the region but I'm not sure how the solid is supposed to look like...

Especially I don't know how they got limits " "! Does represent a hemisphere?

I really appreciate it if anyone could explain these to me.