Question

Finding the length of the curve question.

Answer 1

I am looking at a solution, I marked the area with a red circle that I don't understand. What does it mean when t = 2pi, then x = -2pi ?

Answer 2

\[(-2\pi,1-4\pi^{2},1)\]

This I assume is your point (x,y,z)
I think it is saying that usually you would have 1-t^2=y
y=1-4pi^2

let them equal each otehr

1-t^2=1-4^2
solve for t
t^2=4pi^2
t=2pi, or t=-2pi

but I think they have also made an expression for x in terms of t
which I think is t=-x

so when t=2pi
2pi=-x
-2pi=x

so when t=2pi, x=-2pi (which is the point we want since \[(-2\pi,1-4\pi^{2},1)\]

when t=-2pi

-2pi=-x
2pi=x we dont want this point since it is not on our point \[(-2\pi,1-4\pi^{2},1)\]


this is my best guess, I don't do your topic and don't have your reading material at hand sorry