13th September 2013, 1 min read

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 $a$ .

Torricelli

Volume is

V = π \int_{1}^{a} (\frac{1}{x^{2}}) d x = π (1 - \frac{1}{a})

Surface is

A = 2 π \int_{1}^{a} \frac{1}{x} \sqrt{1 + \frac{1}{x^{4}}} d x \geq 2 π \int_{1}^{a} \frac{1}{x} d x = 2 π \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] );