Definition:Arc Length/Definition 1

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:

$\ds s := \int_a^b \sqrt {1 + \paren {\frac {\d y} {\d x} }^2} \rd x$

Intuition

The arc length of a curve $C$ can be thought of as how long the graph of the function of $C$ would be if cut at the points $\tuple {a, \map f a}$ and $\tuple {b, \map f b}$ and then straightened out.

Also see

• Equivalence of Definitions of Arc Length

• Results about arc length can be found here.

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): arc length
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): 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): arc length
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): length
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): length of an arc