Surface area of a sphere using integrals

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Discussion Overview

The discussion revolves around the derivation of the surface area of a sphere using integrals. Participants explore the method of slicing a hollow sphere into infinitesimal parts and the implications of using different shapes for these slices, specifically comparing cylinders and cones.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about their approach to deriving the surface area of a sphere and seeks assistance in identifying errors in their reasoning.
  • Another participant corrects the first by stating that the slices should be considered as cones rather than cylinders, suggesting that this distinction is crucial for accurate calculations.
  • A later reply discusses the commonality of the question and elaborates on the errors involved in calculating volume and area, highlighting the differences in the nature of these errors when using infinitesimal slices.
  • Participants note that the area error is primarily due to edge effects, while the volume error is related to the distribution of mass within the sphere.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the initial approach to deriving the surface area, as there is a clear disagreement regarding the shapes used for slicing the sphere. The discussion remains unresolved regarding the correctness of the initial method proposed.

Contextual Notes

There are limitations in the assumptions made about the shapes of the slices and the implications of these choices on the calculations. The discussion does not resolve the mathematical steps involved in the integration process.

spec00
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Hello dear colleagues!

Yesterday i was trying to proof the surface area of a sphere formula, then i got some problems. I know that something is seriously wrong in this concept, but i can't tell what exactly is wrong. Could you guys help me please?

I just thougt about a hollow sphere, then we can slice it up to little cylinders with infinitesimal height (like slicing some onion rings). If we add up these teeny weeny little parts, i thought that we could obtain the area of the sphere.

[tex]x^{2}+y^{2}=r^{2}[/tex]
[tex]\int(2*\pi*x)*dy[/tex]

[tex]2*pi*\int\sqrt{y^{2}-r^{2}}*dy[/tex]

But when i integrate over 0 to R and multiply all by 2, the result is not correct. What did i do wrong?

Thanks!
 
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welcome to pf!

hello spec00! welcome to pf! :smile:
spec00 said:
I just thougt about a hollow sphere, then we can slice it up to little cylinders …

no, they're not cylinders (with vertical sides), they're little slices of a cone (with sloping sides), which can have a much larger area :wink:
 
Thanks for the fast reply, tiny-tim!

Well, that's true, but I've seen a proof for the volume of a sphere using slices of it with infinitesimal height. Since the slices are not cylinders, but little cones, that deduction wouldn't get us in the wrong place too?

http://en.wikipedia.org/wiki/Sphere (Proof of the sphere's volume formula)

Thanks again! And sorry about my english, I'm a bit rusty!
 
hi spec00! :smile:
spec00 said:
… I've seen a proof for the volume of a sphere using slices of it with infinitesimal height. Since the slices are not cylinders, but little cones, that deduction wouldn't get us in the wrong place too?

we get this question quite often! …

the error for the volume is the difference between 2πr dh and (roughly) 2π(r + dr/2) dh …

a second-order error of π drdh​

but the error for the area is the difference between 2π dh and 2π dh/cosθ …

a first-order error of 2π(1 - secθ) dh :wink:​

(in layman's terms, most of the volume is in the middle, and the error is only an edge-effect, but for the area, it's all edge! :biggrin:)
 
Wow! Now, that's absolutely clear for me.

Thanks for the huge enlightment..!
 

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