## You are here

Homeequal arc length and area

## Primary tabs

# 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 bounded 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.

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

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 function types 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.

## Mathematics Subject Classification

53A04*no label found*34A34

*no label found*34A05

*no label found*26A09

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections