Question: Find volume of region bounded by torus where equation of torus in spherical coordinates $(r, \theta, \phi)$ is $r=2 \sin \phi$ for $\phi \in [0, \pi]$ and $\theta \in [0, 2\pi]$.
Attempt: Graphing $r=2 \sin \phi$ for $\phi \in [0, \pi]$ shows that $r \in [0, 2]$. Using this as my bounds for integration:
Volume = $\iiint_V 1 dV = \int^\pi_0 \int^{2\pi}_0 \int^2_0 r^2 \sin (\phi) dr d\theta d\phi$ where $r^2 \sin (\phi)$ is the Jacobian for spherical coordinates.
I then evaluated the integral to get an answer for the volume as $\frac{32\pi}{3}$ cubic units, however am unsure of my working to determine the bounds of $r$.
Is this correct?