Originally posted by KribzYes & Yes
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
Originally posted by TheMaster37I 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:
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.
Originally posted by AThousandYoungAs 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.
I think yes and yes.
Originally posted by THUDandBLUNDERGood 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.
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 KribzFinite volume with infinite surface area is possible.
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.