Section 7.5 Polar Arclength (CO5)
Learning Outcomes
Compute arclengths of curves given in polar coordinates.
Activity 7.5.1.
Recall that the length of a parametric curve is given by
\begin{equation*}
\int_{t=a}^{t=b} \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt\text{.}
\end{equation*}
(a)
Let \(x(t)=r\cos(\theta)\) and \(y(t)=r\sin(\theta)\) and show that the length of a polar curve \(r=f(\theta)\) with \(\alpha\leq\theta\leq\beta\) is given by
\begin{equation*}
\int_{\theta=\alpha}^{\theta=\beta} \sqrt{\left(r\right)^2+\left(\frac{dr}{d\theta}\right)^2}d\theta\text{.}
\end{equation*}
(b)
Find an integral computing the arclength of the polar curve defined by \(r=3\cos(\theta)-2\) on \(\pi/3\leq\theta\leq\pi\text{.}\)
(c)
Find the length of the cardioid \(r=1-\cos(\theta)\text{.}\)