Previously in Spheres the coordinates used were chosen to make the integration easy. This section explores using the other alternatives.
Surface Area
The above used this methodology to solve the integral. If you are unfamiliar with it a great summary can be found in PDF here.
Volume of Sphere
I am not sure if the following approach is mathematically rigorous but it satisfies my simple approach.
The integral should have R as its upper limit, not r, as shown below.