= Solution
<Stokes theorem> gives $\int_S(\nabla\times\mathbf F)\cdot\hat{\mathbf n}\,dS=\oint_{\partial S}\mathbf F\cdot d\mathbf r$ for a sufficiently smooth <vector field> and consistently oriented <surface> and boundary. Choose the hemisphere's outward <normal vector>; its boundary circle is then traversed counterclockwise viewed from $+z$. Parametrize it by $\mathbf r=(2\cos t,2\sin t,0)$, $0\leq t\leq2\pi$. On this curve, $\mathbf F=(8\sin t,2\cos t,0)$ and $d\mathbf r=(-2\sin t,2\cos t,0)dt$. Therefore
$$
\boxed{I=\int_0^{2\pi}(-16\sin^2t+4\cos^2t)dt=-12\pi.}
$$
As a check, $\nabla\times\mathbf F=-3\hat{\mathbf z}$, whose flux is $-3$ times the projected disk area $4\pi$, not $-3$ times the hemisphere's surface area. Reversing orientation gives $+12\pi$.
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