# 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

- 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): Entry:**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): Entry:**arc length** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**length**