How Do You Calculate the Area of a Fence Built Over a Curve?

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Discussion Overview

The discussion revolves around calculating the area of a fence built over a curve in a plane, specifically addressing the relationship between the length of the curve and the height of the fence. Participants explore the mathematical justification for the area calculation, considering concepts such as line integrals and integration of functions.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant suggests that the area of the fence can be calculated as the product of the length of the curve (L) and the constant height (H), proposing that the area is LH.
  • Another participant proposes a method of integrating the difference between the upper and lower curves to find the area, asserting that this approach yields a constant area regardless of the curve's shape.
  • A third participant challenges the previous claim by stating that the area calculated using integration does not match the expected area of LH, raising questions about the validity of the integration method presented.
  • Further clarification is sought regarding the need for rigorous justification, specifically whether line integrals are necessary to support the area calculation.

Areas of Agreement / Disagreement

Participants express differing views on the correct method for calculating the area, with some supporting the idea that the area is LH, while others question the integration approach and its implications. The discussion remains unresolved regarding the rigorous justification for the area calculation.

Contextual Notes

Participants highlight potential limitations in their reasoning, including assumptions about the nature of the curve and the integration process, which may affect the area calculation. The discussion does not resolve these limitations.

Castilla
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Think of a rectangular flag. When it is moved by the wind, we may think of the inferior border of the flag as drawing a curve in some plane. But, obviously, the area of the waving flag is the same of the rectangular flag.

So, if I draw a curve of length L in a plane, and I set up a fence of constant height H over said curve, I suppose its area is LH. But what is the rigorous justification?? Line integrals??
 
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The simplest way to do it is to write the lower curve as y= f(x) and the upper curve as y= f(x)+ L. That way, to find the area you are integrating (f(x)+ L)- f(x)= L over the length of the curve. That will be a constant no matter how you bend the curve.
 
Thanks, but if the length of the curve at the base is L and the constant heigth is H, the area of this waving flag, according to your procedure, would be

(Integral) [f(x) + H]dx - (Integral) f(x)dx = (Integral) Hdx.

And that is not LH, which is the obvious area of a flag of length L and heigth H when it is "at rest".
 
Probably I have been unclear with my statement.

I draw a smooth curve in the plane xy. I know its length, it is L. I "build" a fence over it, reaching the same height H for every point of the curve.

It is pretty obvious that this fence has area LH, but what is the rigorous justification? Do I need to go to line integrals to have it?
 

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