There is vector field $F = [x,y,-z]$. We need to find the flux of the vector field outward across the given surface $\sigma=x^2+y^2+z^2=1, x\ge0, y\ge0, z\ge0$ directly and by using Gauss theorem (i.e. $\iiint\limits_V\mathrm{div}\,\mathbf{F}\,dV=\iint\limits_{S}\!\!\!\!\!\!\!\!\!\!\!\;\!\!\;\subset\!\!\supset\mathbf F\cdot\mathbf{n}\,dS$).
In this case the surface is just a sphere.
So, any explanation how to calculate the flux by two ways are highly welcomed. According to Andrei's answer below: $\int_{0}^{\pi}sin\theta d\theta \int_{0}^{2\pi}d\phi=2*2\pi=4\pi$, right?
The normal to the surface is $\mathbf n=[x,y,z]$, where $x^2+y^2+z^2=1$. Then $\mathbf F\cdot\mathbf n=x^2+y^2-z^2=1-2z^2$. Then I would use polar coordinates: $\mathbf F\cdot\mathbf n = 1-2\cos^2\theta$ and $ds=\sin\theta d\theta d\phi$. The integration over $\phi$ is from $0$ to $2\pi$, and the integration of $\theta$ is from $0$ to $\pi$. Can you finish this?
For the divergence part, the integrand is a constant, so you just need to multiply it with the volume of the sphere.