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

$$s={\int }_0^x\sqrt{1+y^{\prime 2}}𝑑x={\int }_0^x\sqrt{1+4a^2{x}^2}𝑑x=\frac{1}{2a}{\int }_0^{2ax}\sqrt{t^2+1}𝑑t.$$

where one has made the substitution (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

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

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

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 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