The shaded region $R$ is bounded by the graphs of the functions $f$ and $g$, where $f ( x ) = x ^ { 2 } - 2 x$ and $g ( x ) = x + \sin ( \pi x )$, as shown in the figure.
(Note: Your calculator should be in radian mode.)
A. Find the area of $R$. Show the setup for your calculations.
B. Region $R$ is the base of a solid. For this solid, at each $x$ the cross section perpendicular to the $x$-axis is a rectangle with height $x$ and base in region $R$. Find the volume of the solid. Show the setup for your calculations.
C. Write, but do not evaluate, an integral expression for the volume of the solid generated when the region $R$ is rotated about the horizontal line $y = - 2$.
D. It can be shown that $g ^ { \prime } ( x ) = 1 + \pi \cos ( \pi x )$. Find the value of $x$, for $0 < x < 1$, at which the line tangent to the graph of $f$ is parallel to the line tangent to the graph of $g$.