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K

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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

K

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Time saving hint: don't try to construct them from scratch. Either you know the answers or you don't. They are intended to be answered Yes or No, not by the construction of f(x).

Kribz

T

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

Gabriel's Trumpet.

T
Kupikupopo!

Out of my mind

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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
Yes & Yes

Acolyte
Now With Added BA

Loughborough

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Originally posted by TheMaster37
Yes & Yes
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:

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.

Nemesio
Ursulakantor

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Originally posted by THUDandBLUNDER
"What about such a function that yields a shape of finite volume but infinite surface area?"

Gabriel's Trumpet.
http://curvebank.calstatela.edu/torricelli/torricelli.htm

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

Cool stuff!

T

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Originally posted by THUDandBLUNDER
"What about such a function that yields a shape of finite volume but infinite surface area?"

Gabriel's Trumpet.
I meant Gabriel's Horn.

AThousandYoung
1st Dan TKD Kukkiwon

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I think yes and yes.

T

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Originally posted by AThousandYoung
I think yes and yes.
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.

K

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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.
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.

cv

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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
Finite volume with infinite surface area is possible.
The simplest example is the Kronecker delta function, rotated about the
x-axis.
The other way round is not possible.

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