= Solution
{c}
Use source <polar coordinates> $x'=s\cos\theta$, $y'=s\sin\theta$ on the <semicircular disk>, with $0\leq s\leq a$ and $0\leq\theta\leq\pi$. Its <surface area element> is $s\,ds\,d\theta$, and the separation from $(0,0,h)$ is $\sqrt{s^2+h^2}$. Thus
$$
\begin{aligned}
\phi(h)
&=-G\sigma\int_0^\pi\int_0^a\frac{s\,ds\,d\theta}{\sqrt{s^2+h^2}}\\
&=-\pi G\sigma\left[\sqrt{s^2+h^2}\right]_0^a
=\boxed{-\pi G\sigma\bigl(\sqrt{a^2+h^2}-h\bigr)},\qquad h>0.
\end{aligned}
$$
This <Newtonian gravitational potential> is half that of the full <disk> because the separation is independent of the polar angle. At large $h$ it approaches $-G(\pi a^2\sigma/2)/h$, with the correct total <mass>.
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