# Homework Help: Volume of a sphere without cap

1. Dec 2, 2013

### thercias

1. The problem statement, all variables and given/known data
A solid sphere of radius R has a spherical cap, defined by the cone theta = alpha, removed from its "north pole". Determine the volume of the sphere without cap.

2. Relevant equations

3. The attempt at a solution
Well obviously, the volume would be volume of sphere - volume of cap.
I am able to derive the volume of the sphere and obtain the formula 4(pi)r^3/3, but I am not sure about the volume of the cap. Is this the same cap that is talked about? http://en.wikipedia.org/wiki/Spherical_cap
If so, the volume would (pi)h^2/3(3r-h), and the answer could be found by subtracting the two quantities. My class focuses on deriving the quantities though, and I am confused and want to understand on where to start for deriving the volume of the cap.

2. Dec 2, 2013

### Simon Bridge

what does "theta=alpha" mean?
Do you mean that the apex angle of the cone is $\alpha$? The half-angle at the apex?

The cap is the regeon described in the wikipedia entry you linked to - yes.
Why not use the same method for the volume of the cap that you used for the volume of the sphere?

For that matter - why not do it in one go by carefully choosing your limits of integration?