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# Torricelli’s trumpet

*Torricelli’s trumpet* is a fictional infinitely long solid of revolution formed when the closed domain

$A:=\{(x,\,y)\in\mathbb{R}^{2}\,\vdots\;\;x\geq 1,\;0\leq y\leq\frac{1}{x}\}$ |

rotates about the $x$-axis. It has a finite volume, $\pi$ volume units, but the area of its surface is infinite; in fact even the area of $A$ is infinite, i.e., the improper integral $\displaystyle\int_{1}^{\infty}\frac{1}{x}\,dx$ is not convergent.

Torricelli’s trumpet is surprising since it can be filled by a finite amount of paint, but this paint can never suffice for painting its surface, no matter how thin a coat of paint is used!

Synonym:

Gabriel's horn

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Definition

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## Mathematics Subject Classification

26A42*no label found*26A36

*no label found*57M20

*no label found*51M04

*no label found*

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