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 $\mathcal C$ 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