Using spherical coordinates, evaluate the triple integral to find the volume inside the sphere: x2 + y2 + z2 = 81 and outside the cylinder x2 + y2 = 49.
You should have the volume element in spherical coordinates somewhere - your textbook or lecture notes. So the only thing you need to set up is the limits of integration. I've found it helps to draw a picture.
So put all that together into a triple integral and show us what you get.
By the way, you can probably do this with a single integral in rectangular coordinates - using slices parallel to the x-y plane. Maybe you could use that to check your answer.....