Calculating Surface area/volume from 2D cross section?

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Discussion Overview

The discussion revolves around calculating the volume of a radially-symmetric 3D object based on its 2D longitudinal cross-section. Participants explore whether it is valid to treat the object as a cylinder with a corresponding cross-sectional area and length, while also considering the implications of different shapes and their volumes.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant suggests that if the cross-sectional area of a radially-symmetric object is known, it might be treated as a cylinder to calculate volume, but acknowledges this may not be universally applicable.
  • Another participant introduces the concept of solids of revolution (SOR) and provides examples of different shapes with the same cross-sectional area but different volumes, questioning the validity of the initial method.
  • A later reply discusses an analogy involving a malleable object, suggesting that if the longitudinal dimension is constrained, the object could be reformed into a cylinder with the same cross-sectional area.
  • One participant challenges the initial hypothesis by presenting a counterexample involving sine and cosine functions, indicating that the hypothesis may be incorrect unless refined.
  • Subsequent replies reflect on the difficulty of proving hypotheses and the potential for intuition to mislead in mathematical reasoning.

Areas of Agreement / Disagreement

Participants express differing views on the validity of using a cylindrical approximation for volume calculations based on cross-sectional area. There is no consensus on whether the initial method is correct, as some participants provide counterexamples that challenge the assumptions made.

Contextual Notes

The discussion highlights limitations in the assumptions made about the shapes and their volumes, as well as the need for further refinement of hypotheses to account for counterexamples.

SuperG
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Calculating Surface area/volume from 2D cross section??

I'm feeling a little stupid today and I need some help...:-p
Assume that I have a radially-symmetric 3D object (for example a candle stick or the base of a table lamp) and that I can calculate the surface area of the largest longitudinal cross section (ie, I split the object precisely along the axis of radial symmety and measure the area of the newly exposed surface). Also assume that the outside profile of this object can be described by a function (though I do not know what the function is)
Is there any reason that I cannot then treat this object as the cross section of a simple cylinder, the product of whose length and diameter is equal to my object's cross sectional surface? In other words, if my object's cross sectional area turned out to be 10 square units with a length of 5), then can I say that my object's volume is equal to the volume of a cylinder with length 5 and diameter 2?
EDIT: corrected an error
 
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You're talking about a solid of revolution. In that case, take note of two different SORs, one cross-section has 2 unit circles each 1 unit symmetrically from the axis, and the other of 2 unit circles each 2 units symmetrically from the axis. The resulting torii have different volumes, but the same cross-sectional area.
For objects which have the axis running through them, consider the unit circle and a square with sides of length \sqrt{\pi}. Then the volume of their SORs are \frac{4\pi}{3} and \frac{\pi^2\sqrt{\pi}}{2} even though their cross-sectional areas were the same.
Also, for cross-sections of equal height and area, consider the absolute values of sine and cosine rotated.
 
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hypermorphism said:
You're talking about a solid of revolution. In that case, take note of two different SORs, one cross-section has 2 unit circles each 1 unit symmetrically from the axis, and the other of 2 unit circles each 2 units symmetrically from the axis. The resulting torii have different volumes, but the same cross-sectional area.

Assuming no hollow center though, would my method work?

I understand that what I'm assuming would not be universally applicable, but may rather be a special case.

I'm going to work on it some more to at least understand where this method fails (FWIW, this is not a class assignment but rather an applied problem--I'm trying to save my lab $3000)

Thanks for your comments
 
Hi Super,
See the rest of my reply.
 
On my drive home I convinced myself that my method is correct (even if my explanation may not have been very good :smile: )

I convinced myself with the following analogy. Imagine that the object in question--lets say it's a turned chair leg with lots of nice curves and a constantly changing diameter--is made not out of wood but out of something like an uncompressible yet malleable clay.
Constraining its longitudinal dimension, it then stands to reason that that chair leg can be reformed into one and only one solid cylinder. The longitudinal cross section of that cylinder will then have an area which is, of course, symmetric around the chair leg's long axis. Obviously this area must then be the same as the area of the chair leg's longitudinal cross section. Therefore, if I can measure (or estimate) the area of the chair leg's longitudinal cross section-because maybe, all I have is a 2D image of the chair leg-I can then come up with a corresponding cylinder (again, holding length constant) which has the same longitudinal cross sectional area and hence, derive a volume for the leg.

Which of course gets me nowhere :rolleyes: because I want to ultimately figure out the surface area of the leg, not the volume. :smile:

Oh well, at least my drive home wasn't boring :smile:
 
The sine and cosine counterexample proves the stated hypothesis wrong, unless you refine the hypothesis further.
 
I'm afraid I see that you are correct.

Just shows that intuition is no match for actually crunching some numbers...

Thanks for the coaxing.
 
No problem. There are a lot of hypotheses that seem like they should be true because they hold for a large amount of examples, but turn out to be false.
 
So how did you see that this was incorrect so quickly? Had you seen this balloon floated before?
 
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I actually didn't see that it was incorrect at first, but I did look for a counterexample. If a nontrivial statement is false, one is sometimes able to find a counterexample faster than finding a proof if a statement is true (ie., why proof by contradiction is so satisfying). If I couldn't find a counterexample, I would've started to see how one would prove it.
I also thought it strange that if it was true, I had never seen it as a cute problem in any of my myriad analysis texts. :smile:
 
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