Question

If we have C: r(t) = (t + t^2 , t^3, t^4) how can we then calculate an approximation of the curve length to C by adding the length of the line connecting 0 to 1, 1 to 2 and 2 to 3?

Answer 1

do you mean evaluate r(t) at t=0, 1, 2, 3 ?
you will get 4 points in 3-dim, with coords (x,y,z)
you can find the distance between any two points using pythagoras (extended to 3-dim)
\[ d = \sqrt{ \Delta x^2 + \Delta y^2+ \Delta z^2}\]

Answer 2

for example, at t=0
r(0)= <0,0,0>
and at t=1
r(1) = <2,1,1>

the distance from r(0) to r(1)= \( \sqrt{ (2-0)^2 + (1-0)^2 + (1-0)^2 } = \sqrt{6} \)