I think my method is correct here, but i must of gone wrong somewhere (note i cannot use divergence theorem)

I have a vector field:

$\displaystyle F = ax\vec{i}$

Where a is a constant and a sphere:

$\displaystyle {x}^{2} + {y}^{2} + {z}^{2} \le {R}^{2}$

R is the radius.

Now i have to do a surface integral i.e. not gauss's divergence theorem of a volume integral. Though i did this in order to match my answers and i got

$\displaystyle \frac{4a\pi {R}^{3}}{3}$

Though i haven't yet found a method to get near to an answer for the surface integral. I know:

$\displaystyle dA = ds cos(\theta)$

$\displaystyle cos(\theta) = \vec{n} . \vec{i or j or k}$

$\displaystyle flux= \phi = \int \vec{F}.d\vec{S} = \iint \vec{F}.\vec{n} dA$

Though in this situation dx, dy and dz are all present:

$\displaystyle \phi = \iint \vec{F}.\vec{n} \frac{dxdy}{\vec{n}.\vec{k}} + \iint \vec{F}.\vec{n} \frac{dxdz}{\vec{n}.\vec{j}} + \iint \vec{F}.\vec{n} \frac{dydz}{\vec{n}.\vec{i}}$

$\displaystyle \vec{N} = \nabla g(x,y,z) = \nabla ({x}^{2} + {y}^{2} + {z}^{2} - {R}^{2}) = 2x\vec{i} + 2y\vec{j} + 2z\vec{k}$

$\displaystyle \vec{n} = \frac{\vec{N}}{|N|}$

then i sub in n into the flux do the dot products, cancel the 1/|N| terms and get:

$\displaystyle \iint \frac{a{x}^{2}}{z} dxdy + \iint ax dydz + \iint \frac{a{x}^{2}}{y} dzdx $

Now i introduce spherical coordinates whereby:

$\displaystyle dx dy = |J1| dr d\theta = 0$

$\displaystyle dy dz = |J2| d\theta d\phi = {r}^{2}{sin}^{2}(\theta) cos(\phi) d\theta d\phi$

$\displaystyle dxdz = |J3| dr d\phi = \frac{r}{2} sin(2\theta)sin(\phi)$

Now it looks a little messy but is still all doable - albeit a little long winded but im not allowed to use the divergence theorem.

To cut it short i get

$\displaystyle 2a\pi{r}^{3} -( \frac{a\pi{r}^{3}}{6} sin(\theta)sin(2\theta))$

Which is not the obvious and the one i got from divergence theorem:

$\displaystyle \frac{4}{3} a\pi {r}^{3}$

Any help is much appreciated and thanks for reading