Answer
$-4 \pi$
Work Step by Step
Stoke's Theorem states that $\iint_S curl F\cdot dS=\int_C F \cdot dr$
We are given that the sphere and the plane $z=1$ intersect on a circle, that is, $z=1; x^2+y^2=4$
Write these in parametric form as: $x=2 \cos t, y=2\sin t; z=t$; $0 \leq t \leq 2 \pi$
Thus, $r=\lt2 \cos t , 2\sin t, t \gt \implies dr=\lt -2\sin t, 2\cos t ,0 \gt$
and $F=x^2yz i+yz^2 j+z^3e^{xy} k \implies F=\lt 8\cos^2 t \sin t,2 \sin t,e^{4 \cos t \sin t} \gt$
Now, $\int_C F \cdot dr=\int_0^{2 \pi} \lt 8\cos^2 t \sin t,2 \sin t,e^{4 \cos t \sin t} \gt\cdot \lt -2\sin t, 2\cos t ,0 \gt dt$
or, $\int_C F \cdot dr=\int_0^{2 \pi}-4 \sin^2 2t+2 \sin 2t dt=\int_0^{2 \pi} 2(\cos 4t-1)+2 \sin 2t dt$
or, $\int_C F \cdot dr=[\dfrac{1}{2}\sin 4 t-\cos 2t-2t]_0^{2 \pi}=-4 \pi$
