Consider an infinitesimal element of area $r d\theta  dr$ which is at a distance $r \sin (\theta)$ from the $x$ axis. 

[![enter image description here][1]][1]

Its moment of inertia is $r d\theta  dr (r \sin (\theta ))^2$.

The moment of inertia about the $x$ axis of the complete sector:

$$ I_x = \int _0^{r_0}\int _{-\frac{\alpha }{2}}^{\frac{\alpha
   }{2}}r^3\sin ^2(\theta) d \theta dr = \frac{1}{8} r_0^4 (\alpha -\sin (\alpha ))$$


  [1]: https://i.sstatic.net/d9NGX.jpg