Hi guys, need some assurance here in evaluating this surface integral.

$\displaystyle \iint_{S} \mathbf{F}\cdot d\mathbf{S}$

where

$\displaystyle \mathbf{F} = \left \langle x^2+y^2+z^2, -2xy+\sin(z), 2z \right \rangle, (x,y,z)\in \mathbb{R}^3$

and

$\displaystyle S:x^2+y^2+(z-5)^2=1$ with positive orientation (outward pointing normal).

Here's what I did:

Using Green's theorem (Divergence theorem)

$\displaystyle \textup{div }\mathbf{F}=2x-2x+2=2$

$\displaystyle \iint_{S} \mathbf{F}\cdot d\mathbf{S}= \iiint_{E} \textup{div }\mathbf{F } dV$

$\displaystyle =2\iiint_{E} dV=2(\textup{volume of sphere})=2( \frac{4}{3} \pi r^3 )=\frac{8\pi}{3}$

Anything wrong with what I did?