Alternative for sphere volume:FAIL

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Discussion Overview

The discussion revolves around an alternative formula proposed for the volume of a sphere, which was derived by a participant's brother. The conversation explores the reasoning behind the formula, its derivation, and the mathematical principles involved, particularly focusing on the role of calculus in understanding the volume of a sphere.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents an alternative formula for the volume of a sphere as (1/2)(∏R²)(2∏R) and questions its validity.
  • Another participant suggests that the formula appears to be based on multiplying the area of a circle by its circumference and expresses curiosity about the reasoning behind this approach.
  • A participant describes the conceptual idea of rotating a circle to form a sphere and notes a potential issue with the varying distances traveled by different parts of the circle during rotation.
  • Integral calculus is mentioned as a necessary tool for summing the volumes generated by the rotation of the circle, highlighting the importance of understanding calculus in this context.
  • Several participants express a lack of knowledge in calculus, with one asking for resources to learn it, while another emphasizes the need to understand differentiation before tackling integration.
  • A later reply acknowledges a participant's realization of their mistake in the reasoning process, indicating a refinement of understanding without resolving the initial question about the formula's validity.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the validity of the alternative formula for the volume of a sphere. There are multiple viewpoints regarding the reasoning behind the formula and the necessity of calculus for proper understanding.

Contextual Notes

Limitations include the lack of a clear derivation of the proposed formula, dependence on understanding calculus concepts, and unresolved questions about the validity of the reasoning presented.

Sulfur
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My brother thought of an alternative formula for the volume of a sphere:
(1/2)(∏R2)(2∏R).
it didn't work

could anyone tell me why?
 
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So it seems to be (area of circle) * (circumference) * (1/2). It's less a question of why it doesn't work, than why your brother thought it would work. Maybe if you posted the derivation, we could point out the problem with it.
 
the idea is that if a circle was turned around 180 degrees on a line through the middle of it, and if every frame of its rotation was kept there, it would make a sphere. The circumference is the distance the edge of the circle has to travel 360 degrees.
hmm... maybe the problem is that different parts of the circle have to travel different lengths to go 360 degrees around. :rolleyes:
Do you think if the problem was solved, I could derive another formula that is correct?
http://imageshack.us/a/img43/9046/circle1.png
http://imageshack.us/a/img839/5682/circle2.png
 
Last edited by a moderator:
This is pretty much integral calculus, summing an infinite number of infinitely small areas. You should learn it, it's really useful.
 
OK
could you give me an integral calculus tutorial? :smile:
 
Sulfur said:
OK
could you give me an integral calculus tutorial? :smile:

To get an idea and basic concepts, try Khan Academy. If you wan to learn the "standard" way, any university calc textbook will do. I'm not sure about online ones, but try Paul's Online Maths Notes
http://tutorial.math.lamar.edu/sitemap.aspx

PS Presuming you've learned differentiation already.
 
i haven't :(
 
Sulfur said:
i haven't :(

Then you should, so you can learn understanding instead of just memorising integrals.
 
Note that if you slowly rotate a circle (in the manner of your pictures), the parts near the edge move farther than the parts near the middle. So every little rotation sweeps out volumes that are greater the farther away from the axis they are. For situations like that, you need calculus (or a smart Greek).

edit: I now noticed that you caught your mistake. well done.
 
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