Prove that the minimum surface of revolution formed by a curve joining two points and is described by revolving the catenary.
You should familiarize yourself with this identity first.
Given the arclength formula , the surface area is thus
Since the integrand is independent of , we can apply the above Euler-Lagrange identity and show that for some arbitrary constant .
With a little algebra, Solving this differential equation we find where is another arbitrary constant. This curve is the famous catenary.
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