Volume of solid inside surface in spherical coordinates.

Question

Find the volume of the solid inside the surface defined by the equation $\rho=8\sin \phi$ in spherical coordinates

So far I've set up an integral in spherical coordinates with $\rho$ from $0$ to $\rho=8\sin \phi$ $\theta$ from $0$ to $2\pi$, $\phi$ from $0$ to $\pi$.

This is just a far off guess as I do not have a clue where to begin. I've been spending a lot of time just looking at the question and trying to make sense of it.

Answer 1

Simply integrate the volume element: $$V= \int_0^{2\pi}\int_0^\pi\int_0^{8\sin\phi}\rho^2\sin\phi\,d\rho d\phi d\theta.$$

Answer 2

Sometimes it helps to have a visual representation of the solid:

It helps you convince yourself that indeed $\phi$ ranges from $0$ to $\pi$ and $\theta$ from $0$ to $2\pi$, as in Martín-Blas Pérez Pinilla 's answer.