arc length of parabolaArcLengthOfParabola2009-06-03 16:32:092009-09-22 11:20:37Examplearc length% this is the default PlanetMath preamble. as your knowledge
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The parabola is one of the quite few plane curves, the arc length of which is expressible in closed form; other ones are line, \PMlinkname{circle}{Circle}, semicubical parabola, \PMlinkname{logarithmic curve}{NaturalLogarithm2}, catenary, tractrix, cycloid, clothoid, astroid, Nielsen's spiral, logarithmic spiral.\, Determining the \PMlinkname{arc length of ellipse}{PerimeterOfEllipse} and hyperbola leads to elliptic integrals.\\
We evaluate the \PMlinkescapetext{length} of the parabola
\begin{align}
y \;=\; ax^2 \qquad (a > 0)
\end{align}
from the apex (the origin) to the point \,$(x,\,ax^2)$.\\
The usual arc length \PMlinkescapetext{formula gives}
$$s \;=\; \int_0^x\!\sqrt{1\!+\!y'^2}\,dx \;=\; \int_0^x\!\sqrt{1\!+4a^2x^2}\,dx
\;=\; \frac{1}{2a}\int_0^{2ax}\!\sqrt{t^2\!+\!1}\,dt.$$
where one has made the \PMlinkname{substitution}{ChangeOfVariableInDefiniteIntegral} $2ax := t$.\, Then one can utilise the result in the entry \PMlinkname{integration of $\sqrt{x^2\!+\!1}$}{IntegrationOfSqrtx21}, whence
\begin{align}
s \;=\; \frac{1}{4a}\left(2ax\sqrt{4a^2x^2\!+\!1}+\arsinh{2ax}\right).
\end{align}
This expression for the parabola arc length becomes especially \PMlinkescapetext{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
$$\frac{1}{4a}(\sqrt{2}+\arsinh{1}) \;=\; \frac{1}{4a}\left(\sqrt{2}+\ln(1\!+\!\sqrt{2})\right).$$