Find the arc length parameterization of $r(t)=\langle e^t\sin t,e^t\cos t,9e^t\rangle$.

Question

I computed $s= \int_{0}^t \! |r'(u)| \, \mathrm{d} u$ to be $s=\sqrt{38e^{2t}}$. Solving for t yields $t=\ln\left(\sqrt{\frac{s^2}{83}}\right)$. However, the system is saying the domain of my function doesn't match that of my answer. Any ideas on what to do?

Answer 1

It seems you have a mistake in evaluating the integral. You have:

$$s = \int^t_0 \sqrt{83e^{2u}}du = \int_0^t\sqrt{83}e^udu = \sqrt{83}e^u\Big|_0^t = \sqrt{83}(e^t-1)$$

Answer 2

Let's try checking your work: $$s=\int_0^t\sqrt{83e^{2u}}du=[\sqrt{83}e^u]_0^t=\sqrt{83}(e^t-1)$$.

Now we solve for $t$: $$t=\ln(\frac s{\sqrt{83}}+1)$$.