# 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

 $\displaystyle\int_{a}^{x}\!\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}\,dx\;=\;% \int_{a}^{x}\!y\,dx.$ (1)

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

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

 $x\!+\!C\;=\;\operatorname{arcosh}{y}.$

Consequently, the equation (2) has the general solution

 $\displaystyle y\;=\;\cosh(x\!+\!C)$ (3)

and the singular solution

 $\displaystyle y\;\equiv\;1.$ (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