- 25 Nov '04 09:18

Yes & Yes*Originally posted by Kribz***Can you construct a function f(x) such that when rotated around the x-axis, it yields a three-dimensional curve whose surface area is finite yet whose volume is infinite?**

What about such a function that yields a shape of finite volume but infinite surface area?

Kribz - 25 Nov '04 11:40 / 1 edit

I get No & Yes. Finite volume, infinite area is fine, but not the other way around. For revolution purposes we may assume that f(x) is non-negative. Then:*Originally posted by TheMaster37***Yes & Yes**

Area = S(2pi*f(x)*sqrt(1+f'(x)^2))dx (where S means integrate over the domain of f)

>= 2pi*S(|f(x)|)dx

So for finite area, f must be 1-integrable.

Volume = pi*S(f(x)^2)dx

So for infinite volume, f must not be 2-integrable.

However, all 1-integrable functions are 2-integrable by Hoelder's inequality in the case where the exponents are both 1/2:

S|fg| =< sqrt(S|f|)sqrt(S|g|)

Hence there is no function for which a finite-area surface and an and infinite-volume solid of revolution exist. - 25 Nov '04 17:25

http://curvebank.calstatela.edu/torricelli/torricelli.htm*Originally posted by THUDandBLUNDER***"What about such a function that yields a shape of finite volume but infinite surface area?"**

Gabriel's Trumpet.

This math is so neat and so inaccessible to me that it blows my mind.

Cool stuff! - 26 Nov '04 20:54

As for objects with infinite volume but finite surface area, I don't think this is possible. One well-known property of a sphere is that it maximizes volume for a given surface area. If infinite volume in finite surface area were possible, then you could improve on a sphere.*Originally posted by AThousandYoung***I think yes and yes.** - 27 Nov '04 17:37 / 1 edit

Good observation. I think this closes this problem, as we have a proof by construction for second question, and a proof by contradiction against the first question.*Originally posted by THUDandBLUNDER***One well-known property of a sphere is that it maximizes volume for a given surface area. If infinite volume in finite surface area were possible, then you could improve on a sphere.** - 03 Dec '04 10:01

Finite volume with infinite surface area is possible.*Originally posted by Kribz***Can you construct a function f(x) such that when rotated around the x-axis, it yields a three-dimensional curve whose surface area is finite yet whose volume is infinite?**

What about such a function that yields a shape of finite volume but infinite surface area?

Kribz

The simplest example is the Kronecker delta function, rotated about the

x-axis.

The other way round is not possible.