equal arc length and area

We want to determine the nonnegative differentiableMathworldPlanetmathPlanetmath real functions  xy  whose graph has the property that the arc lengthMathworldPlanetmath 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

ax1+(dydx)2𝑑x=axy𝑑x. (1)

By the fundamental theorem of calculusMathworldPlanetmathPlanetmath, we infer from (1) the differential equationMathworldPlanetmath

1+(dydx)2=y, (2)

whence  dydx=y2-1.  In the case  y1,  the separation of variablesMathworldPlanetmath yields




Consequently, the equation (2) has the general solution

y=cosh(x+C) (3)

and the singular solution

y 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 parallelMathworldPlanetmathPlanetmath 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