Rotating the region under y = 1/x from 1 to infinity gives Gabriel's horn: finite volume, infinite surface area. You could fill it with paint but never paint it. Where does the paradox dissolve?

A. In the analogy: filling uses volume, painting assumes constant thickness

Both integrals are right and they measure different things: volume converges like 1/x² and surface like 1/x, which diverges. The paint story smuggles in a constant thickness, and far along the horn that coat is thicker than the horn is wide, so the paradox is in the metaphor, not the mathematics.