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Online Notes / Calculus III (Notes) / 3-Dimensional Space / Arc Length with Vector Functions
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In this section we’ll recast an old formula into terms of vector functions.  We want to determine the length of a vector function,


on the interval .


We actually already know how to do this.  Recall that we can write the vector function into the parametric form,




Also, recall that with two dimensional parametric curves the arc length is given by,





There is a natural extension of this to three dimensions.  So, the length of the curve  on the interval  is,




There is a nice simplification that we can make for this.  Notice that the integrand (the function we’re integrating) is nothing more than the magnitude of the tangent vector,




Therefore, the arc length can be written as,



Let’s work a quick example of this.               


Example 1  Determine the length of the curve  on the interval .



We will first need the tangent vector and its magnitude.



The length is then,



We need to take a quick look at another concept here.  We define the arc length function as,



Before we look at why this might be important let’s work a quick example.


Example 2  Determine the arc length function for .



From the previous example we know that,



The arc length function is then,



Okay, just why would we want to do this?  Well let’s take the result of the example above and solve it for t.




Now, taking this and plugging it into the original vector function and we can reparameterize the function into the form, .  For our function this is,




So, why would we want to do this?  Well with the reparameterization we can now tell where we are on the curve after we’ve traveled a distance of s along the curve.  Note as well that we will start the measurement of distance from where we are at .


Example 3  Where on the curve  are we after traveling for a distance of ?


To determine this we need the reparameterization, which we have from above.




Then, to determine where we are all that we need to do is plug in  into this and we’ll get our location.



So, after traveling a distance of  along the curve we are at the point .

Online Notes / Calculus III (Notes) / 3-Dimensional Space / Arc Length with Vector Functions

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