Definition:Arc Length

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Definition

Let $y = f \left({x}\right)$ be a real function which is:

continuous on the closed interval $\left[{a \,.\,.\, b}\right]$

and:

continuously differentiable on the open interval $\left({a \,.\,.\, b}\right)$.


The arc length $s$ of $f$ between $a$ and $b$ is defined as:

$s := \displaystyle \int_a^b \sqrt{1 + \left({\frac {\mathrm dy}{\mathrm dx}}\right)^2}\ \mathrm d x$


Intuition

The arc length of a curve can be thought of as how long the graph of the function would be if cut at the points $\left({a, f \left({a}\right)}\right)$ and $\left({b, f \left({b}\right)}\right)$ and then straightened out.


Also see

For an explanation of this definition and a proof that such an integral exists, see Continuously Differentiable Curve has Finite Arc Length.


Historical Note

The formula for the arc length of a curve was first obtained by Gottfried Wilhelm von Leibniz around $1680$.


Sources