The region bounded by the graph of $y=x^2$, the $x$-axis, and the line $x=2$ is revolved about the line $y=4$. What is the volume of the resulting solid?

Core Concept

Cross-sections are washers with outer radius 4 and inner radius $4-x^2$. The volume is $\pi\int_0^2(16-(4-x^2)^2)\,dx=\pi\int_0^2(8x^2-x^4)\,dx=\pi(\frac{64}{3}-\frac{32}{5})=\frac{224\pi}{15}$.