Reparameterize the curve $r(t)=\langle e^t\sin t,e^t\cos t,5e^t \rangle$ in terms of the arclength parameter, s with $(0,1,5)$ as the base point.

Question

So first, $r^\prime(t)= \langle e^t\sin t+e^t\cos t, e^t\cos t-e^t\sin t,5e^t \rangle$

Then I took the magnitude of $r^\prime(t)$ which is $\sqrt{(e^t\sin t+e^t\cos t)^2+(e^t\cos t-e^t\sin t)^2+(5e^t)^2}$.

Next, I took the integral from $0$ to $t$ of $\sqrt{(e^t\sin t+e^t\cos t)^2+(e^t\cos t-e^t\sin t)^2+(5e^t)^2}dt$ to get $s(t)=3 \sqrt{3}e^t$.

After this I don't know how to get $t(s)$ and then $r(t(s))$.

Answer 1

Watch your integration constant. Since we need $s=t=0$ at the base point, $s=3\sqrt{3}(e^t-1)$. So $t=\ln\left(1+\frac{s}{3\sqrt{3}}\right)$, from which you can get $r$.