# 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 , 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 a coat of paint is used!

Title Torricelli’s trumpet TorricellisTrumpet 2013-03-22 17:17:53 2013-03-22 17:17:53 pahio (2872) pahio (2872) 14 pahio (2872) Definition msc 26A42 msc 26A36 msc 57M20 msc 51M04 Gabriel’s horn