弧长与弧长参数

Length is accumulated speed

Length belongs to the curve, not the walk

Walking at unit speed: the arclength parameter

Which integral gives the length of the curve $\mathbf{r}(t) = (t, t^2, 0)$ for $0 \le t \le 1$?

What is the length of the helix $\mathbf{r}(t) = (\cos t, \sin t, t)$ over $0 \le t \le 2\pi$?

Two parametrizations of the same curve segment are used to compute length. What is the relation between the two results?

The helix $\mathbf{r}(t) = (\cos t, \sin t, t)$ over $0 \le t \le 2\pi$ shadows a full unit circle. Give the ratio of the helix's length to the circle's circumference $2\pi$, as a plain number or expression like sqrt(2).

For the helix $\mathbf{r}(t) = (\cos t, \sin t, t)$ starting at $t = 0$, the arclength function is $s(t) = c \cdot t$. Give the constant $c$, as a plain number or expression like sqrt(2).