32. The Largest Cylinder

A hemisphere of radius $R$ contains a right circular cylinder of radius $r$ and height $h$, whose base lies on the flat face of the hemisphere.

Using integration, derive an expression for the volume of the region inside the hemisphere but outside the cylinder in terms of $r$, $h$, and $R$.

Hence, determine the values of $r$ and $h$ for which the volume of the cylinder is maximized.