Answer
$4.5822$
Work Step by Step
Since, $\iint_S F \cdot dS=\iint_S F \cdot n dS$
where $n$ denotes the unit vector.
and $dS=n dA=\pm \dfrac{(r_u \times r_v)}{|r_u \times r_v|}|r_u \times r_v| dA=\pm (r_u \times r_v) dA$
The flux through a surface can be defined only when the surface is orientable.
Here, we have $\iint_S (x^2+y^2+z^2) dS =\int_{0}^{1} \int_0^1 (x^2+y^2+z^2) \times \sqrt{1+(dx/dt)^2+ (dy/dt)^2} dA$
or, $=\int_{0}^{1} \int_0^1 (x^2+y^2+z^2) \times \sqrt{1+e^{2y}+x^2 e^{2y}} dx dy$
or, $=\int_{0}^{1} \int_0^1 (x^2+y^2+x^2 \times [e^{2y}] ) \times \sqrt{1+e^{2y}+(x^2) \times [e^{2y}]} dx dy$
Now, use a calculating tool.
Hence, we get $ \iint_S (x^2+y^2+z^2) dS \approx 4.5822$
