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