Definition:Arc Length

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:


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

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

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

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