Definition:Arc Length

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:


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

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

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

 * Derivative of Arc Length
 * Arc Length for Parametric Equations
 * Arc Length for Polar Coordinates
 * Arc Length for Vector-Valued Functions

Historical Note
This result was first obtained by around 1680.