Answer
$\approx 4.5822$
Work Step by Step
When the surface is orientable then the flux through a surface is defined as: $\iint_S F \cdot dS=\iint_S F \cdot n dS$
where, $n$ is the unit vector and $dS=n dA=\pm \dfrac{(r_u \times r_v)}{|r_u \times r_v|}|r_u \times r_v| dA=\pm (r_u \times r_v) dA$
Since, $\iint_S (x^2+y^2+z^2) dS =\int_{0}^{1} \int_0^1 (x^2+y^2+z^2) \times \sqrt{1+(\dfrac{dx}{dt})^2+ (\dfrac{dy}{dt})^2} dA$
and $\iint_S (x^2+y^2+z^2) dS =\int_{0}^{1} \int_0^1 (x^2+y^2+z^2) \sqrt{1+e^{2y}+x^2 \times e^{2y}} dx dy=\int_{0}^{1} \int_0^1 (x^2+y^2+x^2 \times e^{2y}) \sqrt{1+e^{2y}+x^2 \times e^{2y}} dx dy$
Need to use calculator tool.
$\iint_S (x^2+y^2+z^2) dS \approx 4.5822$
