Yes it does. The whole region is, in fact, a cylinder with a hemisphere removed, so that there are sharp edges along the unit circle in the xy plane and there is nothing on the z axis.

Actually, I just figured out a great way to think of this without pictures!

If your only concern is the z bounds, they're actually spelled out in that first condition. For all (x,y) z goes between 1 and sqrt(r^2) = r.