Solution
= Solution
A perpendicular <geometric cross-section> at position $x$ is a disk of radius $r=x^2$, so its <area> is $\pi x^4$. Slicing along the axis gives
$$
\boxed{V=\int_0^1\pi x^4dx=\frac\pi5.}
$$
The result uses the coordinate units implicit in the source's relation between radius and axial coordinate.