Use the divergence theorem:
[tex]\displaystyle\iint_{\partial W}\mathbf f\cdot\mathrm d\mathbf S=\iiint_W\nabla\cdot\mathbf f\,\mathrm dV[/tex]
We have divergence
[tex]\nabla\cdot\mathbf f=\dfrac{\partial(y)}{\partial x}+\dfrac{\partial(-4yz)}{\partial y}+\dfrac{\partial(3z^2)}{\partial z}=-4z+6z=2z[/tex]
The volume integral is best computed by first converting to spherical coordinates:
[tex]x=r\cos s\sin t[/tex]
[tex]y=r\sin s\sin t[/tex]
[tex]z=r\cos t[/tex]
Now,
[tex]\displaystyle\iint_{\partial W}\mathbf f\cdot\mathrm d\mathbf S=\int_{t=0}^{t=\pi/2}\int_{s=0}^{s=2\pi}\int_{r=0}^{r=1}2r\cos t\,\mathrm dr\,\mathrm ds\,\mathrm dt=2\pi[/tex]
