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