Gabriel's horn is formed by taking the graph of with the domain and rotating it in three dimensions about the -axis. The discovery was made using Cavalieri's principle before the invention of calculus, but today, calculus can be used to calculate the volume and surface area of the horn between and, where. Using integration, it is possible to find the volume and the surface area : The value can be as large as required, but it can be seen from the equation that the volume of the part of the horn between and will never exceed ; however, it does gradually draw nearer to as increases. Mathematically, the volume approaches as approaches infinity. Using the limit notation of calculus: The surface area formula above gives a lower bound for the area as 2 times the natural logarithm of. There is no upper bound for the natural logarithm of, as approaches infinity. That means, in this case, that the horn has an infinite surface area. That is to say,
When the properties of Gabriel's horn were discovered, the fact that the rotation of an infinitely large section of the -plane about the -axis generates an object of finite volume was considered a paradox. While the section lying in the -plane has an infinite area, any other section parallel to it has a finite area. Thus the volume, being calculated from the "weighted sum" of sections, is finite. Another approach is to treat the horn as a stack of disks with diminishing radii. The sum of the radii produces a harmonic series that goes to infinity. However, the correct calculation is the sum of their squares. Every disk has a radius and an area or. The series diverges but converges. In general, for any real, converges. The apparent paradox formed part of a dispute over the nature of infinity involving many of the key thinkers of the time including Thomas Hobbes, John Wallis and Galileo Galilei. There is a similar phenomenon which applies to lengths and areas in the plane. The area between the curves and from 1 to infinity is finite, but the lengths of the two curves are clearly infinite.
Painter's paradox
Since the horn has finite volume but infinite surface area, there is an apparent paradox that the horn could be filled with a finite quantity of paint and yet that paint would not be sufficient to coat its inner surface. The paradox is resolved by realizing that a finite amount of paint can in fact coat an infinite surface area — it simply needs to get thinner at a fast enough rate. In the case where the horn is filled with paint, this thinning is accomplished by the increasing reduction in diameter of the throat of the horn.
Converse
The converse of Gabriel's horn—a surface of revolution that has a finite surface area but an infinite volume—cannot occur when revolving a continuous function on a closed set:
Since the lateral surface area is finite, the limit superior: Therefore, there exists a such that the supremum is finite. Hence, Finally, the volume: Therefore: if the area is finite, then the volume must also be finite.
