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
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 .