Arc length with vector functions - three dimensional space, Mathematics

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 = ∫ba √ [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 = ∫ba √ [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,

1226_Arc Length with Vector Functions - Three Dimensional Space.png

 Hence, the arc length can be written as,

L = ∫ba || r' (t)|| dt

Posted Date: 4/13/2013 1:31:33 AM | Location : United States

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