Answer
$- \pi$
Work Step by Step
Stokes' Theorem states that $\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr $
The parameterization for the given surface can be written as: $r=\lt \cos t, 0, \sin t \gt \implies dr = \lt - \sin t ,0, \cos t \gt$
Now, $\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr \\= \int_{2 \pi}^{0} \lt 0, \sin t , \cos t \gt \cdot \lt - \sin t ,0, \cos t \gt\\=\int_{2 \pi}^{0} \cos^2 t dt \\=(1/2) \times \int_{2 \pi}^{0} 1+\cos 2t dt \\=\dfrac{1}{2} \times [t+\dfrac{\sin 2t}{t}]_{2 \pi}^0\\ =- \pi$
