Answer
$-18 \pi$
Work Step by Step
Stokes' Theorem states that $\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr $
$=\int_0^{2 \pi} (6 \sin t i+0 j+3 \cos t e^{3 \sin t} k) \cdot (-3 \sin t i + 3\cos t j +0 k) dt$
$=\int_0^{2 \pi} -18 \sin^2 t dt$
Since, $\sin^2 t =\dfrac{1-\cos 2t}{2}$
Now, $\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr=9 \times \int_0^{2 \pi} \cos 2t -1 dt=9 [\dfrac{\sin 2t}{2}-t]_0^{2 \pi} = 9 \times (-2 \pi)=-18 \pi$
