# Property of a Cissoid?

I didn't think it was possible to have a finite area circumscribing an infinite volume but on page 89 of Nonplussed! by Havil (accessible for me at Google Books) it is claimed that such is the goblet-shaped solid generated by revolving the cissoid y$^2$ = x$^3$/(1-x) about the positive y-axis between this axis and the asymptote x = 1. What do you think?

• What exactly are you asking? – Arturo Magidin Jan 25 '11 at 5:35
• I am asking if Havil is mistaken. – ThudanBlunder Jan 25 '11 at 8:12
• @ThudanBlunder: Then perhaps you should actually ask it? As, in the body of your question? – Arturo Magidin Jan 25 '11 at 14:11
• With a small bit of work (involving compact cut-offs and such) and some regularity assumptions on the boundary surface, you can actually show that the claim of a finite area surface bounding an infinite volume violates the isoperimetric inequality. – Willie Wong Jan 25 '11 at 15:08
• @Arturo Magidin: Yes, you are right. I had a few things on my mind and suffered a lapse of concentration. 'What do you think?' is a silly question. – ThudanBlunder Jan 25 '11 at 18:20

Now, as can be seen, they are talking about the cissoid of Diocles, which has the Cartesian equation $y^2=\frac{x^3}{a-x}$ . One can check that the cissoid encloses a finite area along with its asymptote: $\frac34\pi a^2$ ; thus you can expect the corresponding surface of revolution to have finite volume.