equal arc length and area

We want to determine the nonnegative differentiable real functions  $x\mapsto y$  whose graph has the property that the arc length between any two points of it is the same as the area (http://planetmath.org/AreaOfPlaneRegion) by the curve, the $x$-axis and the ordinate lines of those points.

The requirement leads to the equation

${\displaystyle {\int }_a^x}\sqrt{1+{\left({\displaystyle \frac{dy}{dx}}\right)}^2}𝑑x={\displaystyle {\int }_a^x}y𝑑x.$ (1)

By the fundamental theorem of calculus, we infer from (1) the differential equation

$\sqrt{1+{\left({\displaystyle \frac{dy}{dx}}\right)}^2}=y,$ (2)

whence  $\frac{dy}{dx}=\sqrt{y^2-1}$.  In the case  $y\not\equiv 1$,  the separation of variables yields

$$\int 𝑑x=\int \frac{dy}{\sqrt{y^2-1}},$$

i.e.

$$x+C=\mathrm{arcosh}y.$$

Consequently, the equation (2) has the general solution

$y=\cosh (x+C)$ (3)

and the singular solution

$y\equiv \mathrm{\hspace{0.33em}1}.$ (4)

The functions defined by (3) and (4) are the only satisfying the given requirement.  The graphs are a chain curve (which may be translated in the horizontal direction) and a line parallel to the $x$-axis.  Evidently, the line is the envelope of the integral curves given be the general solution.

Title	equal arc length and area
Canonical name	EqualArcLengthAndArea
Date of creation	2013-03-22 19:13:36
Last modified on	2013-03-22 19:13:36
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	8
Author	pahio (2872)
Entry type	Example
Classification	msc 53A04
Classification	msc 34A34
Classification	msc 34A05
Classification	msc 26A09
Synonym	equal area and arc length
Related topic	Arcosh
Related topic	HyperbolicFunctions
Related topic	ChainCurve