Answer
$-4 \pi$
Work Step by Step
Apply Stoke's Theorem $\iint_S curl F\cdot dS=\int_C F \cdot dr$
We need to write these in the parametric form as: $x=2 \cos t, y=2\sin t; z=t$; $0 \leq t \leq 2 \pi$
Then, $r=\lt2 \cos t , 2\sin t, t \gt$;
and $dr=\lt -2\sin t, 2\cos t ,0 \gt$
Now, we have $\int_C F \cdot dr=\int_0^{2 \pi} \lt 8\cos^2 t \sin t,2 \sin t,e^{4 \cos t \sin t} \gt\cdot \lt -2\sin t, 2\cos t ,0 \gt dt$
or, $\int_C F \cdot dr=\int_0^{2 \pi}-4 \sin^2 2t+2 \sin 2t dt$
This implies that
$\int_C F \cdot dr=[\dfrac{1}{2}\sin 4 t-\cos 2t-2t]_0^{2 \pi}=-4 \pi$
