Answer
$\approx 4.5822$
Work Step by Step
The flux through a surface can be defined only when the surface is orientable.
We know that $\iint_S F \cdot dS=\iint_S F \cdot n dS$
Here, $n$ denotes the unit vector.
Since, $\iint_S (x^2+y^2+z^2) dS =\int_{0}^{1} \int_0^1 (x^2+y^2+z^2) \sqrt{1+(dx/dt)^2+ (dy/dt)^2} dA$
$=\int_{0}^{1} \int_0^1 (x^2+y^2+z^2) \sqrt{1+e^{2y}+x^2 e^{2y}} dx dy$
$=\int_{0}^{1} \int_0^1 (x^2+y^2+x^2e^{2y}) \sqrt{1+e^{2y}+x^2 e^{2y}} dx dy$
By using calculator tool, we have
$ \approx 4.5822$