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