Use Gauss Divergence Theorem to comput $$\int \int \limits_S F\cdot n dS$$ where $n$ is the outward normal for the following:
$S$ is the surface of the unit sphere $\{(x,y,z): x^2+y^2+z^2=1\}$, $n$ is the outward normal and $$F(x,y,z)=(xz-z^2,-yz,z+x+y)$$
So if the $div(F)=1$ but what is $V$?
If I make $V=\{x,y \in [0,1], z \in [-\sqrt{1-x^2-y^2},\sqrt{1-x^2-y^2}]\}$, would that work?
V is the whole volume of the ball. Since your divergence is $1$, the integral is just the volume of the unit ball, which is $\frac{4}{3}\pi$.
Using integration, it is
$$\int^1_{-1}\int^{\sqrt{1-x^2}}_{-\sqrt{1-x^2}}\int^{\sqrt{1-x^2-y^2}}_{-\sqrt{1-x^2-y^2}}dzdydx$$
You might have to use trig substitution to calculate it.
Or you can use spherical coordinate,
$$\int^{\pi}_{0}\int^{2\pi}_{0}\int^1_0 r^2\sin{\phi}dr d\theta d\phi$$
which is much easier to calculate.