Answer
$- \pi$
Work Step by Step
The parameterization representation for the given surface can be written as: $r=\lt \cos t, 0, \sin t \gt \implies dr = \lt - \sin t ,0, \cos t \gt$
Our aim is to verify the 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.
$= \int_{2 \pi}^{0} \lt 0, \sin t , \cos t \gt \cdot \lt - \sin t ,0, \cos t \gt$
or, $=\int_{2 \pi}^{0} \cos^2 t dt$
or, $= \int_{2 \pi}^{0} 1+\cos 2t dt \times \dfrac{1}{2} $
or, $=\dfrac{1}{2} \times [t+\dfrac{\sin 2t}{t}]_{2 \pi}^0$
So, $\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr =- \pi$
