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\title{Gabriel's Horn}

\author{Math 166: Honors Calculus II}
\date{Spring 2000}

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``Gabriel's horn'' is the surface generated by revolving the graph of
$y = 1/x$ from 1 to $\infty$ about the $x$-axis.
  \[ \resizebox{3in}{!}{\includegraphics{gabriel.eps}} \]
To calculate the volume of the solid of revolution bounded by the horn,
note that the solid is filled out by infinitesimally thin cylinders
of radius $1/x$ and width $dx$.  So, over any finite interval $[1,a]$
the volume is given by
  \[ \int_1^a \pi \Big(\frac{1}{x}\Big)^2\,dx = \pi\Big(1 - \frac{1}{a}\Big) \]
As $a\to\infty$, the volume approaches $\pi$ which is a resonable
definition of the volume even though the solid has infinite extent.

On the other hand, it can be shown that the surface of Gabriel's horn
over any finite interval $[1,a]$ is given by
\[\int_1^a 2\pi\frac{1}{x}\sqrt{1+\frac{1}{x^4}}\,dx >
\int_1^a 2\pi\frac{1}{x}\,dx = 2\pi\log(a) \]
Since $2\pi\log(a)\to\infty$ as $a\to\infty$, the surface area of
Gabriel's horn cannot be assigned a finite value.  This counter-intuitive
situation could be summarized by saying that Gabriel's horn holds a
finite amount of paint, but not enough to cover its inside surface!

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