The above has to be evaluated by using Divergence theorem.

$\int\int\limits_{S} \bar{N}\bar{F}ds = \int\int\limits_{v}\int \bar{V}\bar{F}dv$

Here

$\bar{F} = 9x\hat{i} + 6y\hat{j} - 10z\hat{k}$

$\bar{V}\bar{F} = \frac{\partial}{\partial x}(9x) + \frac{\partial}{\partial y}(6y) + \frac{\partial}{\partial z}(-10z) = -9 + 6 - 10 = 5$

Now, V is the volume of the sphere of radius 2 and center at the origin.

For integration we use spherical polar coordinates where

$0 \leq r \leq 2, 0 \leq \theta \leq \pi, 0 \leq \phi \leq 2\pi$

And $dv = r^2sin\theta dr d\theta d\phi$

$\therefore \int\int\limits_{v}\int \bar{V}FdV = \int\int\limits_{v}\int 5r^2 sin\theta dr d\theta d\phi \\ \therefore \int\int\limits_{v}\int \bar{V}FdV = 5 \int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\int\limits_{0}^{2} r^2 sin\theta dr d\theta d\phi \\ \therefore \int\int\limits_{v}\int \bar{V}FdV = 5\bigg(\frac{r^3}{3}\bigg)_{0}^{2} (-cos\theta)_{0}^{\pi} (\phi)_{0}^{2\pi} \\ \therefore \int\int\limits_{v}\int \bar{V}FdV = 5*\frac{8}{3}*2*2*\pi \\ \therefore \int\int\limits_{v}\int \bar{V}FdV = \frac{160}{3}\pi$