I am not sure how to get the boundaries for this problem. So far, I have worked out $\mathbf{n}=\nabla(1-r^2-z)=-(2r\mathbf{e}_r-\mathbf{e}_z)$, normalize it so $$\hat{\mathbf{n}}=-\frac{1}{\sqrt{5}}(2r\mathbf{e}_r-\mathbf{e}_z).$$
I think the area element is $dA=rdrd\theta$ and volume element $dV=rdrd\theta dz$ and $$\nabla\cdot\mathbf{v}=\frac{z}{r}.$$
So, I have $$\int_V 1-r^2 drd\theta dz=\int_S -\frac{2}{\sqrt{5}}drd\theta.$$
But I don't know how to find the boundaries for this problem. I'd appreciate any help.
We have $$d\mathbf S = \nabla (z - 1 + r^2) dx dy = (2r \mathbf e_r + z \mathbf e_z) dx dy, \\ z \mathbf e_r \cdot d\mathbf S = 2 r z \,dx dy = 2 r^2 z \,dr d\theta,$$ where $\mathbf e_r$ and $\mathbf e_z$ are associated with cylindrical coordinates. It remains to verify that $$\int_0^1 \int_0^{2 \pi} 2 r^2 z \bigg\rvert_{z = 1 - r^2} d\theta dr = \int_0^1 \int_0^{2 \pi} \int_0^{1 - r^2} \frac z r r \,dz d\theta dr.$$