Arc Length for Parametric Equations
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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): length of an arc