Answer
$$-32 \pi$$
Work Step by Step
Stokes' Theorem states that $\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr$
where, $C$ corresponds to the boundary of the surface oriented counter-clockwise.
We have the boundary of a surface is a circle with parameterization representation as: $r=\lt 4 \cos t, 4 \sin t, 4 \gt$
and $dr = \lt -4 \sin t , 4 \cos t j, 0 \gt$
Now, $F[r(t)]=\lt -4 \sin t , 4 \cos t j, -2k \gt$
Thus, $$\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr =\int_{2 \pi}^{0} \lt -4 \sin t , 4 \cos t j, -2 \gt \cdot \lt -4 \sin t , 4 \cos t ,0 \gt dt \\ \int_{2 \pi}^0 16 \sin^2 t+16 \cos^2 t dt \\=16 \times \int_{2 \pi}^0 dt \\ =16 ( 0-2 \pi) \\=-32 \pi$$
