PlanetMath: Torricelli's trumpet

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (210)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (37)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

Torricelli's trumpet

(Definition)

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

$\displaystyle A := \{(x,\,y)\in\mathbb{R}^2\,\vdots\;\; x \ge 1,\; 0 \le y \le \frac{1}{x}\}$

rotates about the

-axis. It has a finite volume, $\pi$ volume units, but the area of its surface is infinite; in fact even the area of

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!

"Torricelli's trumpet" is owned by pahio. [ full author list (3) ]

(view preamble)

Other names:

Gabriel's horn

This object's parent.

Log in to rate this entry.
(view current ratings)

Cross-references: convergent, improper integral, infinite, surface, area, volume, finite, rotates, domain, closed, solid of revolution
There is 1 reference to this entry.

This is version 11 of Torricelli's trumpet, born on 2007-06-22, modified 2007-07-07.
Object id is 9643, canonical name is TorricellisTrumpet.
Accessed 1180 times total.

Classification:

AMS MSC:	51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries)
	57M20 (Manifolds and cell complexes :: Low-dimensional topology :: Two-dimensional complexes)
	26A36 (Real functions :: Functions of one variable :: Antidifferentiation)
	26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact