Hyperbolic Integral

We will prove:

\(\forall\) \(a\),\(b\) \(\in \mathbb{R}\):

\(\displaystyle \int_{a}^b \sqrt{1+(\frac{d}{dx} \cosh(x))^2}\,dx = \displaystyle \int_{a}^b \cosh(x) \,dx\)

In words: "The arc length of \(\cosh(x)\) in a finite interval is always equal to the area under the curve in the same interval."


To be as complete as possible, we will prove two things first, and the reader will see that the proof for the above will follow directly:

(1) \(\cosh(x) = \cos(ix) \)

(2) \(\sinh(x) = -i\sin(ix)\)

Proof for (1):

\(\cosh(x)\) is defined:

\(\cosh(x) = \frac{e^x + e^{-x}}{2}\)

Let us present an equivalent formulation:

\(\cosh(x) = \frac{ e^{i^4x} + e^{i^2x}}{2}\)

Then by Euler's Formula, we have:

\(\cosh(x) = \frac{1}{2} \left[\cos(i^3x) +i\sin(i^3x) +\cos(ix) + i\sin(ix) \right]\)

\(\Rightarrow\) \(\cosh(x) = \frac{1}{2} \left[ \cos(-ix) + i\sin(-ix) +\cos(ix) +i\sin(ix)\right]\)

\(\Rightarrow\) \(\cosh(x) = \frac{1}{2} \left[ 2\cos(ix) \right] = \cos(ix)\)

This proves (1)

Proof for (2):

Begin with Euler's Formula with a slight modification:

\(e^{i(\phi i)} = \cos(\phi i) + i\sin(\phi i ) \)

\(\Rightarrow\) \(e^{-\phi} -\cos(\phi i) = i\sin(\phi i)\)

\(\Rightarrow\) \(\cos(\phi i) - e^{- \phi} =-i\sin(\phi i)\)

By (1):

\(\Rightarrow\) \(\cosh(\phi) - e^{- \phi} =-i\sin(\phi i)\)

\(\Rightarrow\) \(\frac{e^{\phi} + e^{-\phi}}{2} - e^{-\phi}=-i\sin(\phi i)\)

\(\Rightarrow\) \(\frac{e^{\phi} - e^{-\phi}}{2} = -i\sin(\phi i)\)

\(\Rightarrow\) \(\sinh(\phi) = -i\sin(\phi i)\)

This proves (2).

Now, begin with the arc length integral:

\(\displaystyle \int_{a}^b \sqrt{1+(\frac{d}{dx} \cosh(x))^2}\,dx = \displaystyle \int_{a}^b \sqrt{1+\sinh^2(x)}\,dx\)

By (2):

\( = \displaystyle \int_{a}^b \sqrt{1+(-i\sin(ix))^2}\,dx =\displaystyle \int_{a}^b \sqrt{1-\sin^2(ix)}\,dx \)

\(=\displaystyle \int_{a}^b \sqrt{\cos^2(ix)}\,dx =\displaystyle \int_{a}^b \cos(ix)\,dx \)

And finally, by (1):

\(=\displaystyle \int_{a}^b \cosh(x)\,dx \)


Note by Ethan Robinett
3 years, 8 months ago

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This is one of the interesting properties of the catenary curve, as the only plane curve besides a horizontal line to have this property, i.e., its arc length is proportional to the area under it.

Michael Mendrin - 3 years, 7 months ago

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