Answer
$- \pi$
Work Step by Step
The parameterization for the given surface is: $r=\lt \cos t, 0, \sin t \gt \implies dr = \lt - \sin t ,0, \cos t \gt$
and $F(r(t))=\lt 0, \sin t, \cos t \gt $
Now, $\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr= \int_{2 \pi}^{0} \lt 0, \sin t , \cos t \gt \cdot \lt - \sin t ,0, \cos t \gt$
or, $=\int_{2 \pi}^{0} \cos^2 t dt$
or, $=(1/2) \int_{2 \pi}^{0} 1+\cos 2t dt$
or, $=\dfrac{1}{2} [t+\dfrac{\sin 2t}{t}]_{2 \pi}^0$
Now,$\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr =- \pi$
