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:

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

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

Intuition

The arc length of a curve can be thought of as how long the graph of the curve would be if you cut it at the points $\left(a,f\left({a}\right)\right)$ and $\left(b,f\left({b}\right)\right)$ and then straightened it out.

Also see

• Derivative of Arc Length
• Arc Length for Parametric Equations
• Arc Length for Polar Coordinates
• Arc Length for Vector-Valued Functions

Sources

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