It can be shown that the [[area under the curve]] of the hyperbolic cosine (over a finite interval) is always equal to the arc length (see ''[[Arc_length#Finding_arc_lengths_by_integration|Arcarc length § Finding arc lengths by integration]]'') corresponding to that interval:<ref>{{cite book | title=Golden Integral Calculus | first1=Bali | last1=N.P. | publisher=Firewall Media | year=2005 | isbn=81-7008-169-6 | page=472 | url=https://books.google.com/books?id=hfi2bn2Ly4cC&pg=PA472}}</ref>
<math display="block">\text{area} = \int_a^b \cosh x \,dx = \int_a^b \sqrt{1 + \left(\frac{d}{dx} \cosh x \right)^2} \,dx = \text{arc length.}</math>
since,
<math display="block">\begin{alignat}{0}
& \cosh x = \sqrt{1 + \left(\frac{d}{dx} \cosh x \right)^2}\\
