Arc Length for Polar Curve

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:

and:

We then have:

Also see

 * Arc Length for Parametric Equations