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

[Castilla]

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??

 

[HallsofIvy]

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.

 

[Castilla]

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".

 

[Castilla]

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?