# Torricelli’s Trumpet: Infinite Surface Area but Finite Volume

Just read on Torricelli’s trumpet in Wikipedia. This states that there is a body having infinite surface but finite volume! That sounds contradictory at first.

Function in question is

$\displaystyle{ y=1/x }$

from $x=1$ to $a$.

Volume is

$\displaystyle{ V = \pi \int_1^a \left({1\over x^2}\right)\,dx = \pi\left(1-{1\over a}\right) }$

Surface is

$\displaystyle{ A = 2\pi \int_1^a {1\over x}\sqrt{1 + {1\over x^4}}\,dx \ge 2\pi \int_1^a {1\over x}\,dx = 2\pi\ln a }$

Above plot is from Maxima using:

    plot3d( [x,1/x*sin(v),1/x*cos(v)], [x,1,8], [v,0,2*%pi] );