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# arc length of parabola

The parabola is one of the quite few plane curves, the arc length of which is expressible in closed form; other ones are line, circle, semicubical parabola, logarithmic curve, catenary, tractrix, cycloid, clothoid, astroid, Nielsen’s spiral, logarithmic spiral. Determining the arc length of ellipse and hyperbola leads to elliptic integrals.

We evaluate the length of the parabola

$\displaystyle y\;=\;ax^{2}\qquad(a>0)$ | (1) |

The usual arc length formula gives

$s\;=\;\int_{0}^{x}\!\sqrt{1\!+\!y^{{\prime 2}}}\,dx\;=\;\int_{0}^{x}\!\sqrt{1% \!+4a^{2}x^{2}}\,dx\;=\;\frac{1}{2a}\int_{0}^{{2ax}}\!\sqrt{t^{2}\!+\!1}\,dt.$ |

where one has made the substitution $2ax=:t$. Then one can utilise the result in the entry integration of $\sqrt{x^{2}\!+\!1}$, whence

$\displaystyle s\;=\;\frac{1}{4a}\left(2ax\sqrt{4a^{2}x^{2}\!+\!1}+\arsinh{2ax}% \right).$ | (2) |

This expression for the parabola arc length becomes especially simple when the arc is extended from the apex to the end point $(\frac{1}{2a},\,\frac{1}{4a})$ of the parametre, i.e. the latus rectum; this arc length is

$\frac{1}{4a}(\sqrt{2}+\arsinh{1})\;=\;\frac{1}{4a}\left(\sqrt{2}+\ln(1\!+\!% \sqrt{2})\right).$ |

Here, $\sqrt{2}+\ln(1\!+\!\sqrt{2})=:P$ is called the universal parabolic constant, since it is common to all parabolas; it is the ratio of the arc to the semiparametre. This constant appears also for example in the areas of some surfaces of revolution (see Reese and Sondow).

## Mathematics Subject Classification

53A04*no label found*26A42

*no label found*26A09

*no label found*26A06

*no label found*

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## Recent Activity

new correction: Error in proof of Proposition 2 by alex2907

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## Comments

## Crash of PM system

Hi Administration,

Are all articles made after November lost forever? There are no backups of them? Must the writers write the lost articles anew?

Now it seems that the new system does not allow to edit elder articles.

Jussi

## Re: Crash of PM system

There are no backups. Sadly the last backup was 2011-10-23. They exist in the sense that the authors might (should) still have copies, and copies might exist in various web caches (such as Google).

The editing problem should be fixed now.

apk