info prev up next book cdrom email home

Gabriel's Horn

The Surface of Revolution of the function $y=1/x$ about the $x$-axis for $x\geq 1$. It has Finite Volume

$\displaystyle V$ $\textstyle =$ $\displaystyle \int_1^\infty \pi y^2\,dx = \pi\int_1^\infty {dx\over x^2}$  
  $\textstyle =$ $\displaystyle \pi\left[{-{1\over x}}\right]_1^\infty =\pi[0-(-1)]=\pi,$  

but Infinite Surface Area, since

S&=&\int_1^\infty 2\pi y\sqrt{1+{y'}^2}\,dx\\
&>& 2\pi\int_...
...over x} = 2\pi[\ln x]^\infty_1\\
&=& 2\pi[\ln\infty-0]=\infty.

This leads to the paradoxical consequence that while Gabriel's horn can be filled up with $\pi$ cubic units of paint, an Infinite number of square units of paint are needed to cover its surface!

See also Funnel, Pseudosphere

© 1996-9 Eric W. Weisstein