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'Torricelli's trumpet'
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| Title of object: |
Torricelli's trumpet |
| Canonical Name: |
TorricellisTrumpet |
| Type: |
Definition |
| Created on: |
2007-06-22 11:08:19 |
| Modified on: |
2007-06-22 12:14:22 |
| Classification: |
msc:51M04, msc:57M20, msc:26A36, msc:26A42 |
| Synonyms: |
Torricelli's trumpet=Gabriel's horn |
Revision comment (for changes between this and next version):
solid
The paren is wrong, it ought to be "solid of revolution" |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
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Content:
\emph{Torricelli's trumpet} is a fictional infinitely long surface of revolution formed when the closed domain
$$A := \{(x,\,y)\in\mathbb{R}^2\,\vdots\;\; x \ge 1,\; 0 \le y \le \frac{1}{x}\}$$
rotates about the $x$-axis. It has a finite volume, $\pi$ volume \PMlinkescapetext{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.
The paradoxicality of Torricelli's trumpet may be illustrated by the claim that the body may be filled by a finite amount of paint, but this paint can never suffice for painting the surface of the body, no matter how \PMlinkescapetext{thin} a coat of paint is used!
\PMlinkescapeword{even} |
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