Answer
$-32 \pi$
Work Step by Step
The boundary of the surface is a circle with parameterization:
$r=\lt 4 \cos t, 4 \sin t, 4 \gt \implies dr = \lt -4 \sin t , 4 \cos t j, 0 \gt$
and $F(r(t))=\lt -4 \sin t , 4 \cos t j, -2k \gt$
Stokes' Theorem states that $\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr $
$=\int_{2 \pi}^{0} \lt -4 \sin t , 4 \cos t j, -2 \gt \cdot \lt -4 \sin t , 4 \cos t ,0 \gt dt$
$=\int_{2 \pi}^0 16 \sin^2 t+16 \cos^2 t dt$
$=16 \int_{2 \pi}^0 dt$
$=-32 \pi$
