Torricelli’s trumpet

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

$$A:=\{(x,y)\in {ℝ}^2\mathrm{⋮}x\ge 1,\mathrm{\hspace{0.33em}0}\le y\le \frac{1}{x}\}$$

rotates about the $x$-axis. It has a finite volume, $\pi$ volume , but the area of its surface is infinite; in fact even the area of $A$ is infinite, i.e., the improper integral ${\displaystyle {\int }_1^{\mathrm{\infty }}}{\displaystyle \frac{1}{x}}𝑑x$ 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 a coat of paint is used!

Title	Torricelli’s trumpet
Canonical name	TorricellisTrumpet
Date of creation	2013-03-22 17:17:53
Last modified on	2013-03-22 17:17:53
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	14
Author	pahio (2872)
Entry type	Definition
Classification	msc 26A42
Classification	msc 26A36
Classification	msc 57M20
Classification	msc 51M04
Synonym	Gabriel’s horn