Divergence Theorem: Volume and Surface Integral Solutions

[Phymath]

ok well basically use the divergence thrm on this..
$$\vec{v} = rcos\theta \hat{r} + rsin\theta\hat{\theta} + rsin\theta cos \phi \hat{\phi}$$

so i did (remember spherical coords) i get..
$$5cos \theta - sin\phi$$
taking that over the volume of a hemisphere resting on the xy-plane i get $$10R \pi$$

however with the surface intergral i use $$d\vec{a} = R^3 sin\theta d\theta d\phi \hat{r}$$ where R is the radius of the sphere doing that intergral gives me something obvioiusly R^3 which is not what I'm getting in the volume intergral any help or explanations (if its i need to take the surface of the bottom of the hemi sphere aka the xy-plane in a circle that will suck but let me know!

 

[Reshma]

however with the surface intergral i use $$d\vec{a} = R^3 sin\theta d\theta d\phi \hat{r}$$ where R is the radius of the sphere doing that intergral

Your surface integral is wrong.
$$d\vec{a} = R^2 sin\theta d\theta d\phi \hat{r}$$ not R³.