I have not been able to solve this problem. Can someone please help me find the error?
I think your problem is that $\displaystyle \bigtriangledown \cdot r^2 \hat r $ is not 2r, but rather is 4r. In spherical coordinates you have:
$\displaystyle \bigtriangledown \cdot \vec v = \frac 1 {r^2} \frac {\partial (r^2 v_r)}{\partial r} + \frac 1 {r \sin \theta} \frac {\partial (v_\theta \sin \theta)}{\partial \theta} + \frac 1 {r \sin \theta} \frac {\partial v_\phi}{\partial \phi} $
For the case of $\displaystyle \vec v=r^2 \hat r$ this becomes:
$\displaystyle \bigtriangledown \cdot \vec v = \frac 1 {r^2} \frac {\partial (r^4)}{\partial r} = 4r $
Use this to determine $\displaystyle \int _{Vol} ( \bigtriangledown \cdot \vec v ) dV $ and it turns out to be $\displaystyle 4 \pi R^4$.