[tex]\mathbf f(x,y,z)=e^x\sin y\,\mathbf i+e^x\cos y\,\mathbf j+yz^2\,\mathbf k[/tex]
By the divergence theorem, the surface integral can be evaluated by computing the triple integral
[tex]\displaystyle\iint_Sf(x,y,z)\,\mathrm dS=\iiint_V\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV[/tex]
[tex]=\displaystyle\iiint_V\left(\dfrac{\mathrm d\mathbf f}{\mathrm dx}+\dfrac{\mathrm d\mathbf f}{\mathrm dy}+\dfrac{\mathrm d\mathbf f}{\mathrm dz}\right)\,\mathrm dV[/tex]
[tex]=\displaystyle\int_{x=0}^{x=4}\int_{y=0}^{y=3}\int_{z=0}^{z=4}2yz\,\mathrm dz\,\mathrm dy\,\mathrm dx[/tex]
[tex]=72[/tex]
