I actually have one of my notes online that has pretty much this exact same problem. Attached at the bottem

Go to the definition of spherical co-ordinates. Theta is the angle from the project onto the XY plane. So we want to go from 0-->2pi

Phi is the angle from the z-axis to the line of P. So we want to go from the bound of your cylinder --> pi/2

And for P, well I'll let you look that up in my note because it's best explained there. But generally you see that P must be greater then the line from the origin to the cylinder, so find an equation within the cylinder to model radius with height, and that is your min radius. Of course your max P is that of the sphere. Again, more detail in my notes!

Here you go:

Edit- OOPs, noticed that in my final integral of dV there should be a 2 in front of the intregal, to get both the upper and lower hemispheres!