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).