Answer
$171 \sqrt {14}$
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 $\iint_S x^2 yz dS =\iiint_{D}x^2 y (1+2x+3y) \sqrt {2^2+3^2+1} dA=\iiint_{D}x^2 y (1+2x+3y) \sqrt {4+9+1} dA$
or, $= \sqrt {14} \times \int_{0}^{3} \int_0^2 x^2y+2x^3y+3x^2y^2 dydx$
or, $= \sqrt {14} \times \int_{0}^{3}[\dfrac{x^2y^2}{2}+x^3y^2+x^2y^3]_0^2 dx$
or, $= \sqrt {14} \times \int_{0}^{3} 2x^2 +4x^3+8x^2 dx$
Thus, we have $\iint_S x^2 yz dS=(\sqrt {14}) [\dfrac{10(3)^3}{3}+81)=171 \sqrt {14}$
