Definition:Arc Length

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Definition

Let $y = \map f x$ be a real function which is:

continuous on the closed interval $\closedint a b$

and:

continuously differentiable on the open interval $\openint a b$.


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

$s := \displaystyle \int_a^b \sqrt {1 + \paren {\frac {\d y} {\d x} }^2} \rd x$


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 $\tuple {a, \map f a}$ and $\tuple {b, \map f b}$ 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