Arc Length with Vector Functions
In this part we will recast an old formula into terms of vector functions. We wish to find out the length of a vector function,
r^{→} (t) = {f (t), g(t) , h (t)}
on the interval a ≤ t ≤ b .
in fact we already know how to do this. Remind that we can write the vector function into the parametric form,
x = f (t)
y = g(t)
z = h (t)
As well, remind that with two dimensional parametric curves the arc length is illustrated by,
L = ∫^{b}_{a} √ [f' (t)]^{2} + [g' (t)]^{2} dt
Here is a natural extension of this to three dimensions. Thus, the length of the curve r ?t ? on the interval a ≤ t ≤ b is,
L = ∫^{b}_{a} √ [f' (t)]^{2} + [g' (t)]^{2} + [h' (t)] dt
There is a good simplification which we can make for this.
Note: The integrand that is the function we're integrating is nothing much more than the magnitude of the tangent vector,
Hence, the arc length can be written as,
L = ∫^{b}_{a }|| ^{→}r' (t)|| dt