Horizontal slices of a surface of revolution in spherical coordinates ($dV=\pi [r(\theta )\sin \theta {]}^2|r^{\prime }(\theta )\cos \theta -r(\theta )\sin \theta |d\theta$)

A meridian $r=r(\theta )$ has cylindrical radius $\rho =r\sin \theta$ and height $z=r\cos \theta$. On a branch with monotone height and filled horizontal disk sections, the volume between nearby horizontal planes is $dV=\pi {\rho }^2|dz|$, giving the formula. The conditions matter: a meridian that folds in height or bounds annular sections requires separate branches and inner-radius subtraction. A spherical angular wedge instead has volume $(2\pi /3)r(\theta {)}^3\sin \theta d\theta$ when the solid fills each radial segment. Both describe volumes, but they partition the solid differently.