Let $R$ be the region bounded by the graph of $y = e ^ { 2 x - x ^ { 2 } }$ and the horizontal line $y = 2$, and let $S$ be the region bounded by the graph of $y = e ^ { 2 x - x ^ { 2 } }$ and the horizontal lines $y = 1$ and $y = 2$, as shown above. (a) Find the area of $R$. (b) Find the area of $S$. (c) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when $R$ is rotated about the horizontal line $y = 1$.