A volume sits above the figure in the $xy$-plane bounded by the equations $y = \sqrt{x}$, $y = −x$ for $0 ≤ x ≤ 1$. Each $x$ cross section is a half-circle, with diameter touching the ends of the curves. What is the volume?
a) Sketch the region in the $xy$ plane.
b) What is the area of a cross-section at $x$?
c) Write an integral for the volume.
d) Find the value of the integral.