Answer
$3.4895$
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 $\iint_S x^2y^2 z^2 dS =\iint_{D} x^2 \times y^2 \times z^2 \sqrt{1+(\dfrac{dz}{dx})^2+ (\dfrac{dz}{dy})^2} dA$
or, $=\iint_{D} x^2y^2z^2 \times \sqrt{1+16x^2+ 4y^2} dy dx$
or, $=\iint_{D} x^2y^2 \times (3-2x^2-y^2)^2 \times \sqrt{1+16x^2+ 4y^2} dy dx$
or, $=\int_{-(\sqrt{3/2})}^{(\sqrt{3/2}) } x^2y^2 \times (3-2x^2-y^2)^2 \times \sqrt{1+16x^2+ 4y^2} dy dx$
Now, use a calculating tool.
Hence, we have $\iint_S x^2y^2 z^2 dS \approx 3.4895$
