Arc Length with Vector Functions — Question 6

Question 6 – Solution

Strategy Use half-angle identities and determine the sign of $\sin(t/2)$ on the interval.

Speed $$\vec r'=a\left\langle 1-\cos t,\sin t\right\rangle,\quad \|\vec r'\|=a\sqrt{2-2\cos t}=2a\left|\sin\frac t2\right|.$$ For $0\le t\le 2\pi$, $\sin(t/2)\ge 0$, so the absolute value may be removed.

Length $$L=\int_0^{2\pi}2a\sin\frac t2\,dt =\left[-4a\cos\frac t2\right]_0^{2\pi} =\boxed{8a}.$$

Endpoint note The velocity vanishes at the two cusps, but the speed is continuous and integrable, so the arch still has a well-defined finite length.