Let us assume a>0 and c>0.
Then the we need to consider the following:
1)How much does theta move?
2)What is the radius?
3)What much does phi move?
1)Theta is from 0 to 2pi because it is a complete revolution.
2)The radius is a, because x^2+y^2+z^2=a.
3)We have a cone, the equation of the cone is simply, phi=constant.
But we need to find what that constant is.
We know the equation of cone in rectangular coordinates is,
Convert to Spherical,
rho*sin(phi) = c*rho*cos(phi)
sin(phi) = c*cos(phi)
Look at picture below.
I will do the integral in the next post.