A curious property of catenaries

Suppose you have a flat line f(x) = k and an interval [ab]. Then the area under the line and over the interval is k times the length of the segment of the line.

Surprisingly, the same is true for a catenary with scale k.

With the flat line, the length of the segment of the graph is the same as the length of the segment [ab] on the x-axis, but in general the curve will be longer. The catenary is convex, and it bends so that area under the curve decreases exactly enough balance out the increase in arc length.

The area under a curve f(x) and over the interval [ab] is simply the integral of f from a to b:

\int_a^b f(x)\, dx.

The length of the curve from a to b is also given by an integral:

\int_a^b \sqrt{1 + (f'(x))^2} \, dx.

You can prove the claim above by showing that the first integral is k times the second integral when f(x) = k cosh((xc)/k), the catenary centered at c with scale k.

By the way, this result was discovered independently by Johann Bernoulli and Gottfried Liebnitz three centuries ago.