Let $R$ be the region in the first quadrant bounded by the $y$-axis and the graphs of $y = 4x - x^3 + 1$ and $y = \frac{3}{4}x$.
(a) Find the area of $R$.
(b) Find the volume of the solid generated when $R$ is revolved about the $x$-axis.
(c) Write an expression involving one or more integrals that gives the perimeter of $R$. Do not evaluate.