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

$$V = \int_1^\infty \pi y^2\,dx = \pi\int_1^\infty {dx\over x^2}$$

$$= \pi\left[{-{1\over x}}\right]_1^\infty =\pi[0-(-1)]=\pi,$$

but Infinite Surface Area, since

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

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