I want to compute the surface integral $\int_SF\cdot n \ dS$ with $F(x,y,z)=(x^2+\sin y^2, y^2+\sin z^2, z+\sin (xy))$ and $S$ the surface of the bounded solid that is defined by the surfaces $z=x^2+y^2+1$ and $z=2(x^2+y^2)$ with orientation of the normal to the outside of the solid.
$$$$
We have to compute the surface integral over $S_1:z=x^2+y^2+1$ and $S_2:z=2(x^2+y^2)$ and then we have to add the result, or not?
We have to find the unit normal vector and compute then the dot product with the function $F$, or not?
Can you not use stoke's theorem. The contour that is formed by the two surfaces is a circle ($x^2+y^2=1$ with radius 1 and thus $r = <cost, sint,2>$. Find r'(t) and then get the dot. product of F and r'(t) and then over $2\pi$ evaluate the line integral $\int F.dr$ that will give you the surface integral. But your F looks pretty ugly and it requires a lot of trignometric multiplication and integration.