Arc Length for Polar Curve

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Theorem

Let $a$ and $b$ be real numbers.

Let $\mathcal C$ be a simple curve continuous on $\closedint a b$ and continuously differentiable on $\openint a b$.

Let $\mathcal C$ 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 $\mathcal C$ is given by:

$\displaystyle s = \int_a^b \sqrt {r^2 + \paren {\frac {\d r} {\d \theta} }^2} \rd \theta$


Theorem

Note that $\mathcal C$ satisfies the conditions of Arc Length for Parametric Equations.

So:

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

We have:

\(\displaystyle \frac {\d x} {\d \theta}\) \(=\) \(\displaystyle \frac \d {\d \theta} \paren {r \cos \theta}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\d r} {\d \theta} \cos \theta + r \frac \d {\d \theta} \paren {\cos \theta}\) Product Rule for Derivatives
\(\displaystyle \) \(=\) \(\displaystyle \frac {\d r} {\d \theta} \cos \theta - r \sin \theta\) Derivative of Cosine Function

and:

\(\displaystyle \frac {\d y} {\d \theta}\) \(=\) \(\displaystyle \frac \d {\d \theta} \paren {r \sin \theta}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\d r} {\d \theta} \sin \theta + r \frac \d {\d \theta} \paren {\sin \theta}\) Product Rule for Derivatives
\(\displaystyle \) \(=\) \(\displaystyle \frac {\d r} {\d \theta} \sin \theta + r \cos \theta\) Derivative of Sine Function

We then have:

\(\displaystyle s\) \(=\) \(\displaystyle \int_a^b \sqrt {\paren {\frac {\d r} {\d \theta} \cos \theta - r \sin \theta}^2 + \paren {\frac {\d r} {\d \theta} \sin \theta + r \cos \theta}^2} \rd \theta\)
\(\displaystyle \) \(=\) \(\displaystyle \int_a^b \sqrt {\paren {\frac {\d r} {\d \theta} }^2 \cos^2 \theta - 2 r \frac {\d r} {\d \theta} \sin \theta \cos \theta + r^2 \sin^2 \theta + \paren {\frac {\d r} {\d \theta} }^2 \sin^2 \theta + 2 r \frac {\d r} {\d \theta} \sin \theta \cos \theta + r^2 \cos^2 \theta} \rd \theta\) Square of Sum
\(\displaystyle \) \(=\) \(\displaystyle \int_a^b \sqrt {\paren {\frac {\d r} {\d \theta} }^2 \paren {\sin^2 \theta + \cos^2 \theta} + r^2 \paren {\sin^2 \theta + \cos^2 \theta} } \rd \theta\)
\(\displaystyle \) \(=\) \(\displaystyle \int_a^b \sqrt {r^2 + \paren {\frac {\d r} {\d \theta} }^2} \rd \theta\) Sum of Squares of Sine and Cosine

$\blacksquare$


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