The region in the first quadrant enclosed by the graphs of $y=x$ and $y=x^3$ is revolved about the $x$-axis. What is the volume of the resulting solid?

Core Concept

The curves intersect at $x=0$ and $x=1$, with $x$ on top. The volume is $\pi\int_0^1(x^2-x^6)\,dx=\pi(\frac{1}{3}-\frac{1}{7})=\frac{4\pi}{21}$.