Arc Length — Question 3

Question 3

A bead moves along $$x=3\cos t,\qquad y=3\sin t,\qquad0\le t\le\frac{2\pi}{3}.$$ Find the distance traveled.

Question 3 – Solution

Step 1: Differentiate both coordinates. $$\frac{dx}{dt}=-3\sin t,\qquad\frac{dy}{dt}=3\cos t.$$ Step 2: Compute the speed. $$\begin{align*} \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2} &=\sqrt{9\sin^2t+9\cos^2t}\\ &=\sqrt9=3. \end{align*}$$ Step 3: Integrate the speed. $$\begin{align*} L&=\int_0^{2\pi/3}3\,dt\\ &=[3t]_0^{2\pi/3}=2\pi. \end{align*}$$ $$\boxed{L=2\pi}$$