13th September 2013

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

$$ y = 1 / x $$

from \(x=1\) to \(a\).


Volume is

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

Surface is

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

Above plot is from Maxima using:

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