Definition:Arc Length

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

• 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