Gabriel’s Horn
Rotate the curve y = 1/x around the x-axis from x = 1 to infinity. The resulting trumpet has finite volume — exactly π — but infinite surface area. You can fill it with paint, but you can never paint its surface. Torricelli discovered this paradox in 1643, and it still unsettles intuition.
V = π ∫1∞ (1/x)² dx = π S = 2π ∫1∞ (1/x)√(1 + 1/x&sup4;) dx = ∞
The paradox
Gabriel’s Horn (also called Torricelli’s Trumpet) is formed by rotating the curve y = 1/x around the x-axis for x ≥ 1. The volume integral converges to exactly π, but the surface area integral diverges to infinity. This means you could fill the horn with a finite amount of paint — but that same paint would never be enough to coat the inside surface.
Why the volume converges
The volume uses the disk method: V = π∫(1/x)² dx = π∫1/x² dx. Since 1/x² is a p-series with p = 2 > 1, the integral converges. Evaluated: π[−1/x] from 1 to ∞ = π(0 − (−1)) = π.
Why the surface area diverges
The surface area integral involves 2π∫(1/x)√(1+1/x&sup4;) dx. For large x, the square root approaches 1, so the integral behaves like 2π∫1/x dx = 2π ln(x), which diverges. The surface shrinks too slowly — like the harmonic series.
The painter’s paradox
If you “fill” the horn with π cubic units of paint, the paint touches every point of the interior surface. So in one sense the surface is coated. The resolution is that the paint layer would need to be infinitely thin — zero thickness — which corresponds to zero volume of paint on an infinite area. The paradox exploits the gap between 2D and 3D measures.
History
Evangelista Torricelli published this result in 1643, before the formal development of calculus. He used Cavalieri’s method of indivisibles. The result shocked mathematicians because it demonstrated that infinite length (or area) could bound finite content — a concept that required new mathematical foundations to properly understand.