Let $R$ be the region in the first quadrant enclosed by the graphs of $f(x) = 8x^3$ and $g(x) = \sin(\pi x)$, as shown in the figure.
(a) Write an equation for the line tangent to the graph of $f$ at $x = \frac{1}{2}$.
(b) Find the area of $R$.
(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when $R$ is rotated about the horizontal line $y = 1$.