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Gabriel's horn)
Gabriel's Horn (also called Torricelli's trumpet) is a figure invented by Evangelista Torricelli which has infinite surface area, but finite volume.
Illustration of left hand end of Gabriel's Horn (or Torricelli's trumpet)
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 predates calculus, but is easy to verify by integrating 
and 
. Considering the part of the horn between x = 1 and x = a, the surface area is 2πln(a) while the volume is 
. As a increases, the surface area is unbounded, while the volume is bounded above by π.
At the time it was discovered, this was considered a paradox. The apparent paradox has been described informally by noting that it seems it would take an infinite amount of paint to coat the interior surface, but it also seems that it would be possible to simply fill the interior volume with a finite amount of paint and so coat the interior surface. The resolution of the paradox is that the implication, that an infinite surface area requires an infinite amount of paint, presupposes that a layer of paint is of constant thickness; this is not true in theory in the interior of the horn, and in practice much of the length of the horn is inaccessible to paint, especially where the diameter of the horn is less than that of a paint molecule. - If the paint is considered without thickness, it would further take infinitely long time for the paint to run all the way down to the "end" of the horn.
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