The surface [tex]S[/tex] is the boundary of a cylindrical region [tex]R[/tex] that can be expressed in cylindrical coordinates by the set of points
[tex]\{(r,\theta,z)\mid0\le r\le5\,\land\,0\le\theta\le2\pi\,\land\,0\le z\le7\}[/tex]
By the divergence theorem, the flux of [tex]\vec f[/tex] across [tex]S[/tex] is given by the integral of [tex]\mathrm{div}(\vec f)=-3[/tex] over [tex]R[/tex]:
[tex]\displaystyle\iiint_R-3\,\mathrm dV=-3\int_{z=0}^{z=7}\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=5}r\,\mathrm dr\,\mathrm d\theta\,\mathrm dz[/tex]
[tex]=-21\pi r^2\bigg|_{r=0}^{r=5}=-525\pi[/tex]
