Answer
$0$
Work Step by Step
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.
The boundary of the part of the ellipsoid $4x^2+y^2+4z^2=4$ is a circle $x^2+z^2=1$ and the parameterization of the boundary can be written as: $C: r(t)=\cos ti+0j+\sin t k$ 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 $$\iint_{C} F \cdot dr =\int_0^{2 \pi} (i+e^{(\cos t \sin t)} j+\cos^2 t \sin t k) \cdot (-\sin t i+0 j+\cos t k) dt \\=\int_0^{2 \pi} -\sin t+\cos^3 t \sin t dt \\=\int_{0}^{2\pi} (1-\cos^3 t) \times (-\sin t dt)$$
Consider $cos t =p \implies -\sin t dt =dp$
Thus, $$\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr=\int_{1}^{1} (1-p^3) dp=0$$
