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?
 A: 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.
A: 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").
A: 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.
