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?

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    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
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    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

First, let's have a proper link to the book, alright?

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.

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    To add to this: Havil's interpretation is surely wrong. Look at the quote on the next page from de Sluze: levi opera deducitur mensura vasculi, pondere non magni, quod interim helluo nullus ebibat which he translated as "a drinking glass that has small weight but the heaviest of drinkers cannot empty". The key is the word "weight". A physical goblet has material thickness. So imagine the goblet itself being the volume of revolution formed by the vertical asymptote and the cissoid--it has finite volume and hence finite weight (assuming a constant density material for the goblet). – Willie Wong Jan 25 '11 at 14:51
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    But the amount of liquid it can hold, being the volume of revolution between the y axis and the cissoid, is infinite. Hence a goblet of finite weight (and by rescaling, arbitrarily small weight) but "infinite volume". – Willie Wong Jan 25 '11 at 14:52

The interpretation is mangled. The volume is finite but the surface area infinite, much like Gabriel's Horn. The idea of the quote is much as in Gabriel's Horn: since the volume is finite, you can imagine "filling it up" with a finite amount of paint. But the surface area is infinite, which suggests that you are "painting" an infinite surface with a finite amount of paint, a paradox (of course, quantum mechanics gets in your way, even theoretically). So you get an unbounded area (which de Sluze and Huygens called "infinite") that can be "covered" with a finite quantity (the amount of "paint").

I'm looking at the quote right now, and it must be wrong. The goblet is infinitely tall, with a radius of approximately 1, so it must have infinite area as well. The quote at the end by de Sluze refers to the weight being finite (proportional to the volume under the curve, above the x-axis), while the cup itself can hold an infinite volume. No matter how you play with this, the surface area must be infinite.

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