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