What does this triple integral in spherical coordinate represent?

Question

$$\int_{0}^{a}\int_{0}^{2\pi}\int_{0}^{2\pi} r(b+r \cos\phi) d\phi d\theta dr$$

What does the above integral represent? I don't think it's volume of a figure, since either $\phi$ or $\theta$ should be going from 0 to $\pi$, not 2$\pi$.

Answer 1

It does not depend on $\theta$ anyway so make that a $\int_0^\pi$ and multiply by 2. Factor out from the integrand the Jacobian for volume in spherical. You see something like $\int_0^a \int_0^\pi \int_0^{2\pi} \rho dV$ where $\rho$ is some function of $r$ and $\phi$. Maybe this is a density and you are calculating the mass.