 Solution

The clockwise boundary integral at $z=0$ is
$-{\int }_0^{2\pi }{a}^4({\sin }^4\varphi +{\cos }^4\varphi )d\varphi =-3\pi a^4/2$. Direct integration of the curl over the side and top gives the same value (the $h$-dependent terms cancel), verifying Stokes.