Arc Length of Polar Curves

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The key to computing the length of a polar curve is to think of it as a parametrized curve with parameter $\theta$ . (When computing the slope of a polar curve, we called the parameter $t$ and set $\theta=t$ . Calling the parameter $\theta$ is equivalent and saves a step.) Then

$\begin{eqnarray*}x&=& r \cos(\theta); \cr y &=& r \sin(\theta). \end{eqnarray*}$ Taking derivatives we get

$\begin{eqnarray*}\frac{dx}{d\theta} &=& r' \cos(\theta) - r \sin(\theta); \cr \frac{dy}{d\theta} &=& r' \sin(\theta) + r \cos(\theta), \end{eqnarray*}$ where

$r'$ is shorthand for

$dr/d\theta$ . Squaring gives

$\begin{eqnarray*}\left ( \frac{dx}{d\theta}\right)^2 &=& (r')^2 \cos^2(\theta) + r^2 \sin^2(\theta) - 2 r r' \sin(\theta) \cos(\theta); \cr \left ( \frac{dy}{d\theta}\right)^2 &=& (r')^2 \sin^2(\theta) + r^2 \cos^2(\theta) + 2 r r' \sin(\theta) \cos(\theta). \end{eqnarray*}$ Adding then gives

$\left (\frac{dx}{d\theta}\right )^2 + \left ( \frac{dy}{d\theta} \right )^2 = r^2 + \left ( \frac{dr}{d\theta}\right)^2, \qquad \hbox{so}$

The arc length of a polar curve

$r=f(\theta)$ between

$\theta=a$ and

$\theta=b$ is given by the integral

$L = \int_a^b \sqrt{r^2 + \left ( \frac{dr}{d\theta}\right )^2} \, d\theta.$

In the following video, we derive this formula and use it to compute the arc length of a cardioid.