Use spherical coordinates to find the volume of the region above the cone z = and between the hemispheres z = and z = .
I'm having trouble finding the bounds, would it be
In spherical coordinates, is the distance from the origin, is the angle of the projection on the -plane with the -axis, and is the angle from the -axis. This is relating points to the Cartesian coordinate system, of course.
Our region is a section of a cone between two spheres. Because this cone, defined by
makes an angle of or radians with the -axis, our limits will be . Similarly, our limits will be and our limits will be .
Now, an important thing to consider is that incrementally-defined volume regions in spherical coordinates grow larger as grows larger, and also grow smaller as decreases to (as the volume regions shrink to wedges near the -axis). Nothing like this happens with uniformly distributed Cartesian coordinates. To make up for it, we introduce the factor when calculating triple integrals in spherical coordinates: