= Horizontal slices of a surface of revolution in spherical coordinates
{title2=$dV=\pi[r(\theta)\sin\theta]^2|r'(\theta)\cos\theta-r(\theta)\sin\theta|\,d\theta$}
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.
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