Solution
= Solution
Here $y=0$, the unit <normal vector> is $\mathbf e_y$, and $\mathbf F\cdot\mathbf e_y=x^2$. The square has $-1/2\leq x,z\leq1/2$, so its outward <flux> is
$$
\boxed{\int_{-1/2}^{1/2}\int_{-1/2}^{1/2}x^2\,dx\,dz=\frac1{12}}.
$$
= Solution
Here $y=0$, the unit <normal vector> is $\mathbf e_y$, and $\mathbf F\cdot\mathbf e_y=x^2$. The square has $-1/2\leq x,z\leq1/2$, so its outward <flux> is
$$
\boxed{\int_{-1/2}^{1/2}\int_{-1/2}^{1/2}x^2\,dx\,dz=\frac1{12}}.
$$