The surface box is closed, so we can use the divergence theorem.
[tex]\nabla\cdot\mathbf h=\dfrac{\partial(3xy)}{\partial x}+\dfrac{\partial(z^3)}{\partial y}+\dfrac{\partial(12y)}{\partial z}[/tex]
[tex]\nabla\cdot\mathbf h=3y[/tex]
So the flux is equivalent to the volume integral (denote the space surrounded by [tex]\mathcal S[/tex] by [tex]\mathcal R[/tex])
[tex]\displaystyle\iint_{\mathcal S}\mathbf h\cdot\mathrm d\mathbf S=\iiint_{\mathcal R}\mathrm dV[/tex]
[tex]=\displaystyle\int_{z=0}^{z=7}\int_{y=0}^{y=3}\int_{x=0}^{x=4}3y\,\mathrm dx\,\mathrm dy\,\mathrm dz=378[/tex]
