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Applications of Integration: Arc Length. Dr. Dillon Calculus II Fall 1999 SPSU. Start with something easy. The length of the line segment joining points (x 0 ,y 0 ) and (x 1 ,y 1 ) is. (x 1 ,y 1 ). (x 0 ,y 0 ). The Length of a Polygonal Path?. Add the lengths of the line segments.

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Applications of integration arc length

Applications of Integration: Arc Length

Dr. Dillon

Calculus II

Fall 1999

SPSU


Start with something easy
Start with something easy

The length of the line segment joining points (x0,y0) and (x1,y1) is

(x1,y1)

(x0,y0)


The length of a polygonal path
The Length of a Polygonal Path?

Add the lengths of the line segments.


The length of a curve
The length of a curve?

Approximate by chopping it into polygonal pieces and adding up the lengths of the pieces



What is the approximate length of your curve
What is the approximate length of your curve?

  • Say there are n line segments

    • our example has 18

  • The ith segment connects (xi-1, yi-1) and (xi, yi)

(xi-1,yi-1)

(xi, yi)


The length of that i th segment is
The length of that ith segment is...


The length of the polygonal path is thus
The length of the polygonal path is thus...

which is the approximate length of the curve


What do we do to get the actual length of the curve
What do we do to get the actual length of the curve?

  • The idea is to get the length of the curve in terms of an equation which describes the curve.

  • Note that our approximation improves when we take more polygonal pieces



A basic assumption
A Basic Assumption...

We can always view y as a function of x, at least locally (just looking at one little piece of the curve)

And if you don’t buy that…

we can view x as a function of y when we can’t view y as a function of x...


To keep our discussion simple
To keep our discussion simple...

Assume that y is a function of x

and that y is differentiable

with a continuous derivative


Using the delta notation we now have
Using the delta notation, we now have…

The length of the curve is approximately


Simplify the summands
Simplify the summands...

Factor out

inside the radical to get

And from there



To get the actual arc length l
To get the actual arc length L?

Let

That gives us


What where d you get that
What? Where’d you get that?

Recall that

Where the limit is taken over all partitions

And


In this setting
In this setting...

Playing the role of F(xi) we have

And to make things more interesting

than usual,


What are a and b
What are a and b?

The x coordinates of the endpoints of the arc


Endpoints our arc crossed over itself
Endpoints? Our arc crossed over itself!

One way to deal with that would be to treat the arc in sections.

Find the length of the each section, then add.

a

b


Conclusion
Conclusion?

If a curve is described by y=f(x) on the interval [a,b]

then the length L of the curve is given by


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