Volume of a spherical cap, where h < r The question tells me to find the volume of a spherical cap with height $h$ and radius $r$ (by radius I mean radius of the cap, not the whole sphere).
Well I attempted to solve it, as $R$ is the radius of the sphere, $r =\sqrt{R^2-y^2}$ via Pythagorean theorem. Now $A$ is the cross-section of the sphere, which is $\pi\big(\sqrt{R^2-y^2}\big)^2$. That can be simplified as $\pi(R^2-y^2)$.
Then I integrated it in terms of $R$ and $R-h$ (the height of the spherical cap), and got $\pi(h^2R-h^3/3)$. But how do I express it in terms of $h$ and $r$, where $h < r$?

$$r^2=R^2-(R-h)^2=2Rh-h^2,
\quad\text{that is:}\quad
R={r^2+h^2\over2h}.$$
