# Torricelli’s trumpet

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

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

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 ${\int}_{1}^{\mathrm{\infty}}}{\displaystyle \frac{1}{x}}\mathit{d}x$ 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 |