Applications of Integration: Arc Length

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Applications of Integration Arc Length
Dr. Dillon Calculus II Fall 1999 SPSU
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Start with something easy

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

(x1,y1)
(x0,y0)
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The Length of a Polygonal Path?

• Add the lengths of the line segments.

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The length of a curve?

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

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Approximate the curve with polygonal pieces?
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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)
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The length of that ith segment is...
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The length of the polygonal path is thus...

• which is the approximate length of the curve

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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

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For Ease of Calculation...
Let
and
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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 dont buy that
• we can view x as a function of y when we cant
  view y as a function of x...

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To keep our discussion simple...

• Assume that y is a function of x
• and that y is differentiable
• with a continuous derivative

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Using the delta notation, we now have

• The length of the curve is approximately

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Simplify the summands...

• Factor out

inside the radical to get
And from there
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Now the approximate arc length looks like...
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To get the actual arc length L?

• Let

That gives us
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What? Whered you get that?
Recall that
Where the limit is taken over all partitions
And
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In this setting...

• Playing the role of F(xi) we have

And to make things more interesting than usual,
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What are a and b?

• The x coordinates of the endpoints of the arc

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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
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Conclusion?

• If a curve is described by yf(x) on the interval
  a,b
• then the length L of the curve is given by

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