The integral is like this $$\int \int _Sx(z^2+3x^2)dydz+y(x^2+3y^2)dzdx+z(y^2+3z^2)dxdy$$ where $S:x^2+y^2+z^2=2$
Someone suggested to try with the divergence theorem, but I don't really know how. What i do know is that S is a sphere with normal in (x,y,z) and i need to find a link in the summands but I don't really know how. I can't really see how to form the vector field for this. Thanks for reading, english is not my native language and it is a bit hard to express all this.
The vector-field is $$\vec{F}(\vec{r})=(x(z^2+3x^2), y(x^2+3y^2), z(y^2+3z^2)))^T$$ And by the divergence theorem, we have that $$\int_S \vec{F} \cdot \mathrm{d}\vec{S} = \int_{V} \text{div}(\vec{F)} \mathrm{d}V$$ $$= \int_{V} z^2+9x^2+x^2+9y^2+y^2+9z^2 \mathrm{d}V$$ $$= 10\int_{V} x^2+y^2+z^2 \mathrm{d}V$$ $$= 10\int_{V} r^2 \mathrm{d}V$$ Can you continue?