Infinite Volume but Finite Surface Area

geometry calculus volume

Solution 1:

Aside from my comment about all of $\Bbb R^3$, there is none. A sphere has the smallest surface for a given volume. One pathological case would be fractals that have well defined volume but no well defined surface area. In most cases you would like to say they have infinite area.

Solution 2:

Nope!

See the "Converse" section of http://en.wikipedia.org/wiki/Gabriel's_Horn .

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