[tex]\mathbf F(x,y,z)=x^3\,\mathbf i+2xz^2\,\mathbf j+3y^2z\,\mathbf k[/tex]
The divergence of the vector field is given by
[tex]\nabla\cdot\mathbf F=\dfrac{\partial x^3}{\partial x}+\dfrac{\partial 2xz^2}{\partial y}+\dfrac{\partial 3y^2z}{\partial z}=3x^2+3y^2[/tex]
By the divergence theorem, the integral over the surface [tex]S[/tex] is equivalent to the triple integral over the region bounded by [tex]S[/tex] (call it [tex]R[/tex]).
[tex]\displaystyle\iint_S\mathbf F\cdot\mathrm dS=\iiint_R\nabla\cdot\mathbf F\,\mathrm dV[/tex]
The triple integral can be precisely written as
[tex]\displaystyle\iiint_R\nabla\cdot\mathbf F\,\mathrm dV=3\int_{x=-2}^{x=2}\int_{y=-2}^{y=2}\int_{z=0}^{z=4-x^2-y^2}(x^2+y^2)\,\mathrm dz\,\mathrm dy\,\mathrm dx[/tex]
which is easy enough to evaluate directly. You should find that its value is [tex]\dfrac{512}{15}[/tex].
