Torricelli's Trumpet: Infinite Surface Area but Finite Volume

· klm's blog


Original post is here: eklausmeier.goip.de

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

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] );