Answer
$8 \pi$
Work Step by Step
The parameterization for the given surface can be written as: $r=\lt 2 \cos t, 2 \sin t, 1 \gt $
and $dr = \lt -2 \sin t , 2 \cos t , 0 \gt$
Also, $F(r(t))=\lt -4 \sin t , 2 \sin t , 6 \cos t \gt$
Now, $$\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr \\= \int_{0}^{2 \pi} \lt -4 \sin t , 2 \sin t j, 6 \cos t \gt \cdot \lt -2 \sin t , 2 \cos t j, 0 \gt \\=\int_{0}^{2 \pi} 8 \sin^2 t+4 \cos t \sin t dt \\=\int_{0}^{2 \pi} 8 (\dfrac{1-\cos 2t}{2})+(2)(2) \cos t \sin t dt \\=\int_{0}^{2 \pi} 4+2 \sin 2t -4 \cos 2t dt \\=\int_{0}^{2 \pi} 2 \sin t -8 \cos 2t dt+ \int_0^{2 \pi} 4 dt\\=[4t-2 \sin 2t -\cos 2t]_0^{2 \pi}\\=4 [(2 \pi) -0-1-(0-0-1)] \\=8 \pi$$
