Let $R$ be the region bounded by the curves $y=x, x=1$, and the $x$-axis. $R$ is the base of a solid for which each cross section perpendicular to the $x$-axis is a square. What is the volume of the solid?
$\frac{1}{3}$
2
$\frac{2 \pi}{3}$
$2 \pi$