# Arc Length for Parametric Equations

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## Contents

## Theorem

Let $x = \map f t$ and $y = \map g t$ be real functions of a parameter $t$.

Let these equations describe a curve $\mathcal C$ that is continuous for all $t \in \closedint a b$ and continuously differentiable for all $t \in \openint a b$.

Suppose that the graph of the curve does not intersect itself for any $t \in \openint a b$.

Then the arc length of $\mathcal C$ between $a$ and $b$ is given by:

- $s = \displaystyle \int_a^b \sqrt {\paren {\frac {\d x} {\d t} }^2 + \paren {\frac {\d y} {\d t} }^2} \rd t$

for $\dfrac {\d x} {\d t} \ne 0$

## Proof

\(\displaystyle s\) | \(=\) | \(\displaystyle \int_a^b \sqrt {1 + \paren {\frac {\d y} {\d x} }^2} \rd x\) | Definition of Arc Length | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \int_a^b \sqrt {\paren {\frac {\frac {\d x} {\d t} } {\frac {\d x} {\d t} } }^2 + \paren {\frac {\frac {\d y}{\d t} } {\frac {\d x} {\d t} } }^2} \rd x\) | because $\paren {\dfrac {\frac {\d x} {\d t} } {\frac {\d x}{\d t} } }^2 = 1$, and from corollary to chain rule | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \int_a^b \sqrt {\paren {\frac {\d x} {\d t} }^2 + \paren {\frac {\d y} {\d t} }^2} \paren {\frac 1 {\frac {\d x} {\d t} } } \rd x\) | factoring $\dfrac {\d x}{\d t}$ out of the radicand. No absolute value is needed as length cannot be negative. | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \int_a^b \sqrt {\paren {\frac {\d x} {\d t} }^2 + \paren {\frac {\d y} {\d t} }^2} \frac {\d t} {\d x} \rd x\) | Derivative of Inverse Function | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \int_a^b \sqrt {\paren {\frac {\d x} {\d t} }^2 + \paren {\frac {\d y} {\d t} }^2} \rd t\) | Integration by Substitution |

$\blacksquare$

## Also see

## Sources

- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards:
*Calculus*(8th ed.): $\S 10.3$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Leonardo of Pisa**