Area of a fence over a curve

In summary, when considering a rectangular flag, the area of the flag remains the same even when it is moved by the wind, as the inferior border of the flag draws a curve in some plane. To find the area of the waving flag, one can use the formula LH, where L is the length of the curve and H is the constant height of the fence built over the curve. This can also be justified using line integrals.
  • #1
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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  • #2
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.
 
  • #3
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".
 
  • #4
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?
 

1. What is the area of a fence over a curve?

The area of a fence over a curve is the total surface area that is enclosed by the fence when it is placed over a curved boundary. This can be visualized as the space inside the fence when viewed from above.

2. How is the area of a fence over a curve calculated?

To calculate the area of a fence over a curve, you need to divide the curved boundary into small, straight segments. Then, calculate the area of each segment using basic geometry formulas such as the area of a rectangle or triangle. Finally, add up the areas of all the segments to get the total area of the fence over the curve.

3. Can the area of a fence over a curve be calculated using calculus?

Yes, the area of a fence over a curve can be calculated using calculus. This method involves taking the limit of the sum of areas of smaller and smaller segments as the size of the segments approaches zero. This results in an integral that can be solved to find the total area of the fence over the curve.

4. How does the shape of the curve affect the area of the fence?

The shape of the curve can significantly affect the area of the fence. A curved boundary with a larger radius will result in a larger area compared to a boundary with a smaller radius. Additionally, the shape of the curve can also determine the number and size of the segments needed for calculation, which can impact the overall accuracy of the area calculation.

5. Are there any practical applications of calculating the area of a fence over a curve?

Yes, there are several practical applications of calculating the area of a fence over a curve, such as in landscaping or construction projects. Knowing the total area of a fence over a curved boundary can help in determining the amount of materials needed, or in estimating costs for a project. It can also be used in designing structures that require curved boundaries, such as curved walls or domes.

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