Find the time it takes a body to travel $8$ meters in the curve $r(t)=(\cos t, \sin t, t)$ from $t=0$

Question

Given the curve $r(t)=(\cos t, \sin t, t)$ and that a body starts moving from $t=0$, after how much time will the body travel the distance of $8$ meters?

It seems I need to use the definition of curve length: $$ \int_a^b \sqrt{(x'(t))^2+(y'(t))^2+(z'(t))^2} $$

In my case I'm not sure if I'm choosing the right interval: $$ \int_0^8 \sqrt{(-\sin^2t)^2+\cos^2t+1}=\int_0^8 \sqrt{2}=8\sqrt{2} $$

Answer 1

The upper limit is time dependent, you need to find a $t'$ such that,

$$\int_{0}^{t'} \|\gamma'(t)\| \ dt = 8$$

Answer 2

You're doing the problem backwards. Given a time $t_1$, the distance traveled between $t = 0$ and $t = t_1$ is given by $$ \int_0^{t_1} \|r'(t)\|\,dt = \sqrt{2} t_1 $$ What you're looking for is the time $t_1$ such that $8 = \sqrt{2} t_1$.

Answer 3

$\int_{0}^{t_0} \|\gamma'(t)\| \ dt = 8 \Leftrightarrow \int_0^{t_0} \sqrt{\sin^2t +\cos^2t +1} = 8 \Leftrightarrow \sqrt{2}t_0 = 8 \Leftrightarrow t_0 = 4\sqrt{2}$