# 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:

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

## 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

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

- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards:
*Calculus*(8th ed.): $\S 7.4$