Arc Length for Parametric Equations

Theorem

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

Let these equations describe a curve $\CC$ 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 $\CC$ between $a$ and $b$ is given by:

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

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

Proof

$\ds s$

$=$

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

Definition of Arc Length

$\ds $

$=$

$\ds \int_a^b \sqrt {\paren {\frac {\frac {\d x} {\d t} } {\frac {\d x} {\d t} } }^2 + \paren {\frac {\frac {\d y}{\d t} } {\frac {\d x} {\d t} } }^2} \rd x$

because $\paren {\dfrac {\frac {\d x} {\d t} } {\frac {\d x}{\d t} } }^2 = 1$, and from corollary to chain rule

$\ds $

$=$

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

factoring $\dfrac {\d x} {\d t}$ out of the radicand. No absolute value is needed as length cannot be negative.

$\ds $

$=$

$\ds \int_a^b \sqrt {\paren {\frac {\d x} {\d t} }^2 + \paren {\frac {\d y} {\d t} }^2} \frac {\d t} {\d x} \rd x$

Derivative of Inverse Function

$\ds $

$=$

$\ds \int_a^b \sqrt {\paren {\frac {\d x} {\d t} }^2 + \paren {\frac {\d y} {\d t} }^2} \rd t$

Integration by Substitution

$\blacksquare$

Also see

Sources

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.): $\S 10.3$
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): arc length
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): length of an arc
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): arc length