Gabriel's horn — finite volume, infinite surface area

What just happened?

By rotating the curve $y = 1/x$ (for $x \ge 1$) about the X axis, you get an infinitely long horn — Gabriel’s horn. Astonishingly: although the solid is infinitely long, its volume is finite and equals exactly $\pi$, yet its surface area is infinite.

Volume — a convergent integral:

The integral $\int_1^\infty 1/x^2\,dx$ is convergent — the volume tends to the finite value $\pi \approx 3.14$.

Surface area — a divergent integral:

The integral $\int_1^\infty 1/x\,dx$ is divergent (it grows like $\ln x$, exactly like the harmonic series) — the surface area has no limit at all.

The painter’s paradox:

Hence the famous question: you can fill the horn with a finite amount of paint (volume $\pi$), but the same paint would not be enough to paint its infinite inner surface. The resolution: “filling” and “painting” are different notions (volume vs area), and the paint layer has a thickness that also tends to zero as the horn narrows.

Torricelli’s discovery (1643):

The paradox was discovered by Evangelista Torricelli — even before the full development of differential calculus! He was astonished by his own result. It is a classic example of the difference between a convergent and a divergent integral and of how infinity breaks physical intuition.

Why you need this for the exam:

Gabriel’s horn is a flagship example of solids of revolution and improper integrals. Trick: you compute both quantities for the same solid, but check separately whether each integral converges. $\int 1/x^2$ converges, $\int 1/x$ diverges — which is why a finite volume and an infinite surface area coexist.

Remember: Gabriel’s horn has a finite volume $\pi$ and an infinite surface area — because $\int_1^\infty 1/x^2$ converges while $\int_1^\infty 1/x$ diverges. A finite volume does not guarantee a finite surface area.