I read the wiki article and it turns out the horn uses precisely that dimensional difference in a very clever way. The radius varies as 1/x along the x-axis, so

I read the wiki article and it turns out the horn uses precisely that dimensional difference in a very clever way. The radius varies as 1/x along the x-axis, so the volume grows as 1/x^2 while the surface area grows as 1/x. This means the growth rate of volume goes to zero much faster than that of the surface area. As a result the volume integral converges, while the surface area diverges.