Arc Length of Right Parabolic Segment

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Theorem

Let $ABC$ be a right parabolic segment where:

$AB$ is the defining chord $\LL$ of $ABC$
$C$ is the vertex of the defining parabola $\PP$ of $ABC$.


Area-of-Right-Parabolic-Segment.png


The length $\LL$ of the arc $ACB$ is given by:

\(\ds \LL\) \(=\) \(\ds \dfrac {\sqrt {b^2 + 16 a^2} } 2 + \dfrac {b^2} {8 a} \map \ln {\dfrac {4 a + \sqrt {b^2 + 16 a^2} } b}\)
\(\ds \) \(=\) \(\ds \dfrac {\sqrt {b^2 + 16 a^2} } 2 + \dfrac {b^2} {2 a} \inv \sinh {\dfrac {2 a} b}\)


where:

$a$ is the length of the line segment $CF$, where $F$ is the point at which the axis of $\PP$ intersects $AB$
$b$ is the length of the line segment $AB$.


Proof


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Sources

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