Answer
$12$
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, $dS=n dA=\pm \dfrac{(r_u \times r_v)}{|r_u \times r_v|}|r_u \times r_v| dA=\pm (r_u \times r_v) dA$
$\iint_{S_3} (z+x^2y) dS=\iint_D [(\sin u+(v)^2 \cos u )] du dv$
$=\int_{0}^{3} \int_0^{\pi/2} \sin u+v^2 \cos u du dv$
$=\int_{0}^{3} [-\cos u +v^2 \sin u]_0^{\pi/2} dv$
$=\int_{0}^{3} v^2+1 dv$
$=[v^3/3+v]_{0}^{3}$
$=12$
