Answer
$$0$$
Work Step by Step
Stokes' Theorem can be defined as: $\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr $
Te 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, $F(r(t))=i+e^{\cos t \sin t} j+\cos^2 t \sin t k$
$$\iint_{C} F \cdot dr =\int_0^{2 \pi} (i+e^{\cos t \sin t} j+\cos^2 t \sin t k) \cdot (-\sin t i+0 j+\cos t k) dt\\=\int_0^{2 \pi} -sin t+\cos^3 t \sin t dt \\=\int_{0}^{2\pi} (1-\cos^3 t) (-sin t dt)$$
Consider $\cos t =u $ and $-\sin t dt=-du$
Now, $$\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr=\iint_{C} F \cdot dr\\ =\int_{1}^{1} (1-u^3) du \\=0 $$
