Question

Evaluate s(t)=∫t−[infinity]||r′(u)||du for the bernoulli spiral r(t)=⟨etcos(8t),etsin(8t)⟩. It is convenient to take −[infinity] as the lower limit since s(−[infinity])=0. Then use s to obtain an arc length parametrization of r(t).

Answer 1

$\vec r(t)=\langle e^t\cos8t,e^t\sin8t\rangle$

$\|\vec r'(t)\|=\sqrt{(e^t(\cos8t-8\sin8t))^2+(e^t(\sin8t+8\cos8t))^2}=e^t\sqrt{(\cos8t-8\sin8t)^2+(\sin8t+8\cos8t)^2}$

$\implies\|\vec r'(t)\|=e^t\sqrt{65}$

Then

$s(t)=\displaystyle\sqrt{65}\int_{-\infty}^te^u\,\mathrm du=\sqrt{65}e^t$