Mathematical Definition
Gabriel's horn is formed by taking the graph of, with the domain (thus avoiding the asymptote at x = 0) and rotating it in three dimensions about the x-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 x = 1 and x = a, where a > 1. Using integration (see Solid of revolution and Surface of revolution for details), it is possible to find the volume and the surface area :
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 will get closer and closer to as becomes larger. Mathematically, the volume approaches as approaches infinity. Using the limit notation of calculus, the volume may be expressed as:
This is so because as approaches infinity, approaches zero. This means the volume approaches (1 - 0) which equals .
As for the area, the above shows that the area is greater than times the natural logarithm of . There is no upper bound for the natural logarithm of as it approaches infinity. That means, in this case, that the horn has an infinite surface area. That is to say;
- as
or
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