Evaluate the surface integral $\int_S \mathbf{F} \cdot \ d \mathbf{S}$ where $\mathbf{F}=2xy\boldsymbol{i}+3x^{2}\boldsymbol{j}+2yz\boldsymbol{k}\ $ and $S$ is the surface $z=xe^y,\ 0 \le x \le 1, 0 \le y \le 1$, with upwards orientation.

$\int\int_S \mathbf{F} \cdot \ d \mathbf{S}=$