Answer
$0$
Work Step by Step
Stokes' Theorem states that $\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr $
The boundary of the part of the ellipsoid $4x^2+y^2+4z^2=4$ is a circle $x^2+z^2=1$
The parameterization of the boundary is: $C: r(t)=\cos ti+0j+\sin t k$ and $dr=(-sin ti+0j +\cos t k) dt$
Now, $F(r(t))=i+e^{\cos t \sin t} j+\cos^2 t \sin t k$
and $\iint_{C} F \cdot dr =\int_0^{2 \pi} (i+e^{\cos t \sin t} j+\cos^2 t \sin t k) \cdot (-\sin t i+0 j+\cos t k) dt$
or, $=\int_0^{2 \pi} -sin t+\cos^3 t \sin t dt$
or, $=\int_{0}^{2\pi} (1-\cos^3 t) (-sin t dt)$
Let us suppose that, $u=\cos t $ and $du =-\sin t dt$
Therefore, $\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr=\int_{1}^{1} (1-u^3) du=0$
