Solution
= Solution
The <volume form> in <cylindrical coordinates> is $dV=r\,dr\,d\phi\,dz$. Integrating around the axis and over the height gives cylindrical-shell <volume> $2\pi hr\,dr$. Hence the <volume integral> is
$$
\boxed{K=\pi h\int_0^a r\,v(r)^2\,dr
=\pi h\int_0^a r^3\omega(r)^2\,dr}.
$$
For unit <density>, this is the <kinetic energy> of the rotational motion.