Arc Length for Polar Curve
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Theorem
Let $a$ and $b$ be real numbers.
Let $\CC$ be a simple curve continuous on $\closedint a b$ and continuously differentiable on $\openint a b$.
Let $\CC$ be described by the parametric equations:
- $\begin {cases} x & = r \cos \theta \\ y & = r \sin \theta \end {cases}$
where:
- $r$ is a function of $\theta$
- $\theta \in \closedint a b$.
Then the length $s$ of $\CC$ is given by:
- $\ds s = \int_a^b \sqrt {r^2 + \paren {\frac {\d r} {\d \theta} }^2} \rd \theta$
Theorem
Note that $\CC$ satisfies the conditions of Arc Length for Parametric Equations.
So:
- $\ds s = \int_a^b \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta$
We have:
\(\ds \frac {\d x} {\d \theta}\) | \(=\) | \(\ds \frac \d {\d \theta} \paren {r \cos \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\d r} {\d \theta} \cos \theta + r \frac \d {\d \theta} \paren {\cos \theta}\) | Product Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\d r} {\d \theta} \cos \theta - r \sin \theta\) | Derivative of Cosine Function |
and:
\(\ds \frac {\d y} {\d \theta}\) | \(=\) | \(\ds \frac \d {\d \theta} \paren {r \sin \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\d r} {\d \theta} \sin \theta + r \frac \d {\d \theta} \paren {\sin \theta}\) | Product Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\d r} {\d \theta} \sin \theta + r \cos \theta\) | Derivative of Sine Function |
We then have:
\(\ds s\) | \(=\) | \(\ds \int_a^b \sqrt {\paren {\frac {\d r} {\d \theta} \cos \theta - r \sin \theta}^2 + \paren {\frac {\d r} {\d \theta} \sin \theta + r \cos \theta}^2} \rd \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_a^b \sqrt {\paren {\frac {\d r} {\d \theta} }^2 \cos^2 \theta - 2 r \frac {\d r} {\d \theta} \sin \theta \cos \theta + r^2 \sin^2 \theta + \paren {\frac {\d r} {\d \theta} }^2 \sin^2 \theta + 2 r \frac {\d r} {\d \theta} \sin \theta \cos \theta + r^2 \cos^2 \theta} \rd \theta\) | Square of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_a^b \sqrt {\paren {\frac {\d r} {\d \theta} }^2 \paren {\sin^2 \theta + \cos^2 \theta} + r^2 \paren {\sin^2 \theta + \cos^2 \theta} } \rd \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_a^b \sqrt {r^2 + \paren {\frac {\d r} {\d \theta} }^2} \rd \theta\) | Sum of Squares of Sine and Cosine |
$\blacksquare$