# Definition:Arc Length

Jump to navigation
Jump to search

It has been suggested that this page or section be merged into Definition:Contour/Length.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Mergeto}}` from the code. |

It has been suggested that this page or section be merged into Definition:Length of Curve.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Mergeto}}` from the code. |

## 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.

## Historical Note

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

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**arc length** - 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards:
*Calculus*(8th ed.): $\S 7.4$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**arc length** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**length** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**length of an arc**