Let $R$ be the region in the first and second quadrants bounded above by the graph of $y = \frac{20}{1 + x^{2}}$ and below by the horizontal line $y = 2$.
(a) Find the area of $R$.
(b) Find the volume of the solid generated when $R$ is rotated about the $x$-axis.
(c) The region $R$ is the base of a solid. For this solid, the cross sections perpendicular to the $x$-axis are semicircles. Find the volume of this solid.