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 equilateral triangles. What is the volume of the solid?

Core Concept

Each triangle has side $1-x^2$, so its area is $\frac{\sqrt{3}}{4}(1-x^2)^2$. The volume is $\frac{\sqrt{3}}{4}\int_{-1}^1(1-x^2)^2\,dx=\frac{\sqrt{3}}{4}\cdot\frac{16}{15}=\frac{4\sqrt{3}}{15}$.