Why Do Sphere Surface Areas Differ With Same Approximation?

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Discussion Overview

The discussion revolves around the calculation of the surface area of a sphere using different approximations, specifically addressing the discrepancies that arise in the results obtained. Participants are examining the mathematical reasoning behind their calculations and the assumptions made in the process.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant claims to have derived the surface area of a sphere using two methods but obtained different results, prompting a request for clarification.
  • Another participant challenges the assumption that transforming a frustum into a cylinder is valid, suggesting that this leads to an incorrect conclusion.
  • A participant defends their approach by stating that they assumed the radii are equal in both methods, which they believe justifies the transformation from frustum to cylinder.
  • Concerns are raised about a specific mathematical step where an extra sine term appears in the area differential, questioning its validity and origin.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are competing views regarding the validity of the assumptions made in the calculations and the transformations applied.

Contextual Notes

There are unresolved mathematical steps and assumptions regarding the transformation of geometric shapes that may affect the accuracy of the calculations presented.

joseph_seb
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i have found the area of sphere in two ways using the same approximation.but i get two different answers ;one the correct value 4∏R^2?how does this happen?
i'm attaching the solution below?please refer to the attachments and give a solution?
 

Attachments

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  • spherearea2.jpg
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Your mistake is where "transform the frustrum to a cylinder". You don't actually do a "tranformation" you just assume frustrum is equal to the cylinder and that is not true.
 


You write dA = 2Pirh and h = rsin(theta)dtheta, but then you write dA = 2Pir^2 sin^2(theta) dtheta. Where does the extra sin(theta) come from?
 

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