Arc Length for a Helix
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Find the arc length, L, of the helix
Equation 1
for
.
How do we proceed?
We need to find the length of the tangent vector. Usually this depends on the parameter t.
Then what?
We integrate over the interval specified.
State that mathematically.
Equation 2
How do we find the tangent vector?
We find it by differentiating the equation of the position vector from Equation 1.
Set that up.
Can we apply the differentiation to each component separately?
Yes.
Show that.
Do the differentiation.
We get
How do we evaluate the length of this vector?
The length equals the square root of the sum of the squares of the components.
Set that up.
Evaluate this.
Using the trig identity, we get
Equation 3
This is an unusual case. Usually the tangent vector is a function of t.
Combine Equations 2 and 3.
We get
.
Let's check this result by graphing.
What is the shape of this curve?
The trig dependence suggests a circle.
What about the y-component?
It increases linearly.
What is the net effect?
A helix.
How is it oriented?
Along the y-axis.
Sketch this helix.
Diagr1
Let's plot the point for
. How do we determine its coordinates?
We substitute
into Equation 1.
Do that.
Plot this point on your paper graph.
Diagram2 Note that the coordinates are shown in color.
Do the same for
Plot this point on your paper graph.
Diagram3 Note that the coordinates are shown in color.
Now we can see the portion of the curve specified. Let's estimate its length by comparing the scale indicated with the arc length. That is, compare the distance along the y axis from -5 to 5 (about 1 inch) with the distance along the curve between the two points.
It looks like the distance along the curve is four to five times longer.
How many graph units is that?
40 to 50, in reasonable agreement with our calculation.
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor