I'm trying to solve this problem:
Compute the volume of the solid bounded by:
- the surface $(z+1)^2=x^2+y^2,$
- the surface $4z=x^2+y^2,$
- above the $xy$ plane.
I want to do it with spherical coordinates. However $\rho$ is difficult to express because it's no longer $\sqrt{x^2+y^2+z^2}$. $\rho$ should be a function of $\phi$ (the angle between the positive semi-axis $z$ and a ray $\overrightarrow{(0,0,0)(x,y,z)}$). $\phi$ will vary from $\arcsin \left(\dfrac{1}{\sqrt{5}}\right) $ to $\dfrac{\pi}{2}$. And $\rho(\phi)$ varies from $1$ to $\sqrt{5}$.
Any ideas of how to do it by spherical coordinates? I appreciate your help.
You can always cast the problem in spherical polar coordinates, but there is a discontinuity in the gradient of $\rho(\phi)$ along the circle that is the intersection of the surfaces $(z+1)^2 = x^2 + y^2$ and $4z = x^2 + y^2$. This will needlessly complicate the volume integral.
If I were allowed to, I'd use cylindrical polar coordinates instead of spherical, because of the cylindrical symmetry of the volume in question.