The base of a solid is the region bounded by the graph of $y=1-x^2$ and the $x$-axis. Cross sections perpendicular to the $x$-axis are semicircles with diameters in the base. What is the volume of the solid?

Core Concept

Each semicircle has diameter $1-x^2$, so its area is $\frac{1}{2}\pi(\frac{1-x^2}{2})^2=\frac{\pi}{8}(1-x^2)^2$. The volume is $\frac{\pi}{8}\int_{-1}^1(1-x^2)^2\,dx=\frac{\pi}{8}\cdot\frac{16}{15}=\frac{2\pi}{15}$.