Alternative for sphere volume:FAIL

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SUMMARY

The discussion centers around an incorrect alternative formula for the volume of a sphere proposed by a user, which is expressed as (1/2)(∏R²)(2∏R). The formula fails because it does not account for the varying distances traveled by different parts of the circle during rotation, which is crucial for deriving the volume of a sphere. Integral calculus is identified as the necessary mathematical tool to correctly derive the volume of a sphere, emphasizing the importance of understanding differentiation before tackling integration. Resources such as Khan Academy and Paul's Online Math Notes are recommended for learning these concepts.

PREREQUISITES
  • Understanding of basic geometry, specifically the properties of circles and spheres.
  • Familiarity with integral calculus concepts.
  • Knowledge of differentiation and its applications.
  • Ability to interpret mathematical formulas and derivations.
NEXT STEPS
  • Study integral calculus to understand volume derivations.
  • Learn differentiation techniques to grasp foundational calculus concepts.
  • Explore the relationship between rotation and volume in calculus.
  • Review resources like Khan Academy and Paul's Online Math Notes for structured learning.
USEFUL FOR

Students of mathematics, educators teaching calculus, and anyone interested in understanding the geometric principles behind the volume of three-dimensional shapes.

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
 
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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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