 Conical flux by closing with a cap

To evaluate outward flux through the lateral surface of a finite right circular cone, add its planar cap and apply the divergence theorem to the enclosed solid. Subtract the cap flux, whose outward normal points away from the solid. For a cone $r=az$ with $0\le z\le h$, the volume element is $rdrd\phi dz$ and the cap has radius $ah$. The cone's vertex has no unique normal, but is a set of area zero; one can remove a small tip and take a limit when the field is regular there.