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.): $\S 7.4$
