Paul's Online Math Notes
Calculus III (Notes) / 3-Dimensional Space / Arc Length with Vector Functions   [Notes] [Practice Problems] [Assignment Problems]


On August 21 I am planning to perform a major update to the site. I can't give a specific time in which the update will happen other than probably sometime between 6:30 a.m. and 8:00 a.m. (Central Time, USA). There is a very small chance that a prior commitment will interfere with this and if so the update will be rescheduled for a later date.

I have spent the better part of the last year or so rebuilding the site from the ground up and the result should (hopefully) lead to quicker load times for the pages and for a better experience on mobile platforms. For the most part the update should be seamless for you with a couple of potential exceptions. I have tried to set things up so that there should be next to no down time on the site. However, if you are the site right as the update happens there is a small possibility that you will get a "server not found" type of error for a few seconds before the new site starts being served. In addition, the first couple of pages will take some time to load as the site comes online. Page load time should decrease significantly once things get up and running however.

August 7, 2018

Calculus III - Notes
  Next Chapter Partial Derivatives
Tangent, Normal and Binormal Vectors Previous Section   Next Section Curvature

 Arc Length with Vector Functions

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 .

Tangent, Normal and Binormal Vectors Previous Section   Next Section Curvature
  Next Chapter Partial Derivatives

Calculus III (Notes) / 3-Dimensional Space / Arc Length with Vector Functions    [Notes] [Practice Problems] [Assignment Problems]

© 2003 - 2018 Paul Dawkins