This question is based a little on orbit mechanics, so the sphere has the Earth at the centre (but isn't the Earth) - not relevant but helps me explain it.
Given a hollow sphere I can calculate the volume as such:
V = 4/3*pi*(R^3-r^3)
Where: V = volume, R = outer radius, r = inner radius
Now suppose I want to calculate the volume of a segment of this hollow sphere. The segment is defined by two angles of inclination from the N-S direction. On a 2D plane these angle draw lines from the centre of the Earth to the outer edge of the hollow sphere, and have are symmetrical around the N-S axis. Here's a picture to explain:
Here the red lines are the defined by the 2 angles of inclination, the blue lines are the projection of the red lines into 3 dimension.
My issue here is I can't work out how to calculate the volume of this halo structure. At first I thought I could calculate it from 2 spherical caps (larger - small), but that isn't right because a spherical cap is cut horizontally, and these cuts are clearly at an angle. Is there a name for this structure that I should be using in my research?
In spherical coordinates the region is described as $$ r\le\rho\le R,\quad \phi_1\le\phi\le\phi_2,\quad 0\le\theta\le2\,\pi. $$ It's volume is then $$ \int_r^R\int_0^{2\pi}\int_{\phi_1}^{\phi_2}\rho^2\sin\phi\,d\phi\,d\theta\,d\rho. $$