Question

What is the arc length of `r(t) = (t^2, 1-t^2, -t)` in the interval `[0,1]`?

Answer 1

`s =1.81161`

Explanation:

Considering `r(t)=(x(t),y(t),z(t))` we know that
`ds^2=dx^2+dy^2+dz^2` or
`((ds)/(dt))^2=((dx)/(dt))^2+((dy)/(dt))^2+((dz)/(dt))^2`
Also that
`s=int_0^1 ((ds)/(dt))dt = int_0^1 (sqrt(((dx)/(dt))^2+((dy)/(dt))^2+((dz)/(dt))^2))dt` so
`s=int_0^1 ((ds)/(dt))dt = int_0^1 (sqrt((2t)^2+(-2t)^2+(-1)²))dt`
then
`s = int_0^1(sqrt(8 t^2+1))dt`
`s = 1/8 (4 t sqrt(1 + 8 t^2) + sqrt(2)times"arcsinh"(2 sqrt(2) t))|_0^1 = 1.81161`