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.

Torricelli

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] );
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One thought on “Torricelli’s Trumpet: Infinite Surface Area but Finite Volume

  1. Pingback: Statistics of this Blog: 2.600 Views, 78 Countries, 20 Comments | Elmar Klausmeier's Weblog

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