Solution
= Solution
The <moment of inertia> about an axis is
$$
I=\int_V\rho\,r_\perp^2\,dV,
$$
where $r_\perp$ is the perpendicular distance to the axis. The disc mass is
$$
M=\rho\pi r^2\delta.
$$
To leading order in $\delta/r$, the distance from the $x$-axis is $|y|$. Using polar coordinates in the disc,
$$
\begin{aligned}
I_x
&=\rho\delta\int_0^r\int_0^{2\pi}
(s\sin\phi)^2s\,d\phi\,ds\\
&=\rho\delta\frac{r^4}{4}\pi
=\boxed{\frac14Mr^2}.
\end{aligned}
$$