= Solution
{c}
The solid <cylinder> has constant <mass density> $\rho=m/(\pi a^2l)$. A thin cylindrical shell of radius $s$ and thickness $ds$ has <mass> $dm=2\pi\rho ls\,ds$. Its contribution to the axial <moment of inertia> is $s^2dm$. Therefore
$$
\mathcal I_{33}=\int_0^a s^2(2\pi\rho ls)\,ds=2\pi\rho l\frac{a^4}{4}=\boxed{\frac12ma^2.}
$$
The attached wheels are massless, so they add no <moment of inertia>. This is the <moment of inertia of a uniform solid cylinder> about its central longitudinal <axis of rotation>.
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