TorricellisTrumpet 2007-06-22 11:08:19 2007-07-07 09:14:52 Definition surface of revolution % 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} \PMlinkescapeword{even} \emph{Torricelli's trumpet} is a fictional infinitely long solid of revolution formed when the closed domain \[ A := \{(x,\,y)\in\mathbb{R}^2\,\vdots\;\; x \ge 1,\; 0 \le y \le \frac{1}{x}\} \] 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. 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 \PMlinkescapetext{thin} a coat of paint is used!