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

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

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

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

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

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

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