# 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 (http://planetmath.org/Circle), semicubical parabola^{}, logarithmic curve (http://planetmath.org/NaturalLogarithm2), catenary^{}, tractrix^{}, cycloid^{}, clothoid^{}, astroid, Nielsen’s spiral, logarithmic spiral^{}. Determining the arc length of ellipse^{} (http://planetmath.org/PerimeterOfEllipse) and hyperbola^{} leads to elliptic integrals^{}.

We evaluate the of the parabola

$y=a{x}^{2}\mathit{\hspace{1em}\hspace{1em}}(a>0)$ | (1) |

from the apex (the origin) to the point $(x,a{x}^{2})$.

The usual arc length

$$s={\int}_{0}^{x}\sqrt{1+{y}^{\prime 2}}\mathit{d}x={\int}_{0}^{x}\sqrt{1+4{a}^{2}{x}^{2}}\mathit{d}x=\frac{1}{2a}{\int}_{0}^{2ax}\sqrt{{t}^{2}+1}\mathit{d}t.$$ |

where one has made the substitution (http://planetmath.org/ChangeOfVariableInDefiniteIntegral) $2ax=:t$. Then one can utilise the result in the entry integration of $\sqrt{{x}^{2}+1}$ (http://planetmath.org/IntegrationOfSqrtx21), whence

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

This expression for the parabola arc length becomes especially 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

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

Here, $\sqrt{2}+\mathrm{ln}(1+\sqrt{2})=:P$ 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 http://mathworld.wolfram.com/UniversalParabolicConstant.htmlReese and Sondow).

Title | arc length of parabola |

Canonical name | ArcLengthOfParabola |

Date of creation | 2013-03-22 18:57:19 |

Last modified on | 2013-03-22 18:57:19 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 13 |

Author | pahio (2872) |

Entry type | Example |

Classification | msc 53A04 |

Classification | msc 26A42 |

Classification | msc 26A09 |

Classification | msc 26A06 |

Synonym | closed-form arc lengths |

Related topic | FamousCurvesInThePlane |

Related topic | AreaFunctions |

Defines | universal parabolic constant |