Surface Area Equality: The Simplest Explanation

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Discussion Overview

The discussion revolves around the equality of two surface area expressions related to a function's surface area in the context of surfaces of rotation. Participants seek clarification on the derivation and validity of the formulas presented, as well as the appropriate mathematical framework for calculating surface area.

Discussion Character

  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant presents two surface area expressions and requests a simple explanation for their equality.
  • Another participant questions the origin of the formulas and points out that surface area calculations typically involve surface integrals in 3D space, suggesting that the provided formulas resemble those used for arc length calculations.
  • A third participant expresses confusion over the initial formulas, questioning their validity and whether integrals should be included, while also inquiring if the discussion pertains to surfaces of rotation.
  • The original poster acknowledges the context of rotation but admits to not including integral signs in their initial query.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the validity or derivation of the surface area formulas. There is confusion and disagreement regarding the appropriate mathematical approach to the problem.

Contextual Notes

There is a lack of clarity regarding the assumptions behind the formulas presented, and the discussion does not resolve the mathematical steps necessary for a complete understanding of the surface area calculations.

calculushelp
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ok why does

surface area = ( y) ( 1+ f ' (x)^(2) )^(1/2)



equal


surface area = (x ) (1+ f ' (x)^(2) )^(1/2)



the simplest explanation please.
 
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Where did you get those formulae from? And surface area of what? You're supposed to do a surface integral in 3D space (the details of which depend on the parametrisation of the surface area)to get the surface area, but I don't see any integral sign. The formulae you gave resembles the one given for arc length calculation.

For a surface area of a function in 3D, for which the surface z(x,y) is given, the surface area of the portion that projects down onto a region R is given by [tex]\int \int_R \sqrt{\left(\frac{\partial z}{\partial x} \right)^2 + \left(\frac{\partial z}{\partial y}\right)^2 + 1} \ \ dA[/tex]
 
calculushelp said:
ok why does

surface area = ( y) ( 1+ f ' (x)^(2) )^(1/2)



equal


surface area = (x ) (1+ f ' (x)^(2) )^(1/2)



the simplest explanation please.

There can be no explanation- what you have written makes no sense at all! Are there supposed to be integrals in there?
Are you talking about the area of surfaces of rotation?
 
yes! in rotation! sorry i didnt know how to put an integral sign.
 

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