Opposite of Gabriel's Horn Paradox

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Discussion Overview

The discussion revolves around the mathematical properties of integrals related to the concept of Gabriel's Horn, particularly exploring the idea of whether an infinite area can yield a finite volume of revolution and vice versa. Participants examine specific integrals and their implications in the context of this paradox.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • The original poster (OP) presents an integral ∫ from 0 to 1 1/x2/3 dx, claiming it evaluates to 3, and questions whether the converse of Gabriel's Horn can hold true when considering the volume of revolution.
  • Some participants reference Gabriel's Horn as an example of the Painter's Paradox, asserting that an infinite volume with a finite area is not possible.
  • One participant suggests that the OP may be miscalculating the surface area or volume of the proposed function.
  • The OP clarifies that by "area," they meant "area under the curve," distinguishing it from the surface area of the solid of revolution.
  • Another participant expresses agreement with the OP's observation, acknowledging the insight shared.

Areas of Agreement / Disagreement

There is no consensus on whether the converse of Gabriel's Horn can be true. Some participants challenge the OP's calculations and interpretations, while others support the OP's findings, indicating a mix of agreement and disagreement.

Contextual Notes

Participants note potential confusion between different definitions of area, specifically distinguishing between area under the curve and surface area of solids of revolution. There are unresolved questions regarding the calculations and implications of the integrals discussed.

fakecop
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We are all familiar with Gabriel's Horn, where the function f(x) = 1/x generates an infinite area but a finite volume when revolved around the x-axis.

So the other day I stumbled upon a particular interesting integral: ∫ from 0 to 1 1/x2/3 dx

Instead of infinite limits of integration, we have an infinite integrand. evaluating this integral, we have 3.

When we rotate the integral about the x-axis, however, we have pi* ∫ from 0 to 1 1/x4/3 dx, which diverges to infinity.

Is this possible? I know that an infinite area can produce a finite volume of revolution, but can the converse of the statement be true? or have I done something wrong?
 
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SteamKing said:
http://en.wikipedia.org/wiki/Gabriel's_Horn

Gabriel's horn is an example of the Painter's Paradox. According to the article above, the converse (infinite volume, finite area) is not true.

I don't think you are properly calculating the surface area or the volume of the your example function.

See: http://en.wikipedia.org/wiki/Pappus's_centroid_theorem

I think the OP is using "area" in the sense of "area under the curve", not "surface area of the solid of revolution", which is what the Painter's Paradox is concerned with.
 
Yes, sorry for not clarifying but by area I meant "area under the curve", which is a different calculation than "surface area of the solid of revolution". With solids of revolution that are infinitely long, I cannot find a connection between the surface area and the area under the curve.
 
Very interesting observation. I believe you are correct in your finding. Thank you for sharing this insight.

Junaid
 

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