Answer
$4.92429$
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.
Since, $\iint_S xyz dS =\iint_{D} xyz \sqrt{1+(dz/dx)^2+ (dz/dy)^2} dA$
$=\iint_{D} xyz \sqrt{1+(2xy^2)^2+(2x^2y)^2} dx dy$
$=\int_0^2 \int_0^1 xyz \sqrt{1+4x^2y^4+ 4x^4y^2} dx dy$
$=\int_0^2 \int_0^1 x^3y^3 \sqrt{1+4x^2y^4+ 4x^4y^2} dx dy$
Now use a calculating tool.
$ \approx 4.92429$
