By the divergence theorem, the flux of [tex]\mathbf f(x,y,z)=4x\,\mathbf i+xy\,\mathbf j+5xz\,\mathbf k[/tex] across the boundary [tex]\partial E[/tex] of the cube [tex]E[/tex] is equal to the integral of [tex]\nabla\cdot\mathbf f[/tex] over [tex]E[/tex]:
[tex]\displaystyle\iint_{\partial E}\mathbf f\cdot\mathrm d\mathbf S=\iiint_E\nabla\cdot\mathbf f\,\mathrm dV[/tex]
We have divergence
[tex]\nabla\cdot\mathbf f=\dfrac{\partial(4x)}{\partial x}+\dfrac{\partial(xy)}{\partial y}+\dfrac{\partial(5xz)}{\partial z}=4+6x[/tex]
and so the flux is
[tex]\displaystyle\iiint_E(4+6x)\,\mathrm dV=\int_{z=0}^{z=2}\int_{y=0}^{y=2}\int_{x=0}^{x=2}(4+6x)\,\mathrm dx\,\mathrm dy\,\mathrm dz[/tex]
[tex]=4(4x+3x^2)\bigg|_{x=0}^{x=2}=80[/tex]
