Answer
$$0$$
Work Step by Step
The surface is the part of the circle $x^2+y^2=1 $ .
The parameterization of the boundary is: $C: r(t)=\cos t i+\sin t j+0 k \implies dr=(-\sin t i+\cos t j+0k) dt$
We have: $F(r(t))=\cos^2 t \sin (0) i+\sin^2 t j+\cos t \sin t k=0 i i+\sin^2 t j+\cos t \sin t k$
Stokes' Theorem states that $\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr $
$=\int_0^{2 \pi} (0 i+\sin^2 t j+\cos t \sin t k) \cdot (-\sin t i+ \cos t j+0 k) dt$
$=\int_{0}^{2 \pi} \sin^2 t \cos t dt$
Plug in $a=\sin t $ and $da =\cos t dt$
Now, $\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr=\int_0^{2 \pi} a^2 da=0$
