1573 measuring arclength
TRANSCRIPT
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Applications of Integration: Arc Length
Dr. DillonCalculus IIFall 1999
SPSU
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Start with something easy
The length of the line segment joining points (x0,y0) and (x1,y1) is
210
210 )()( yyxx
(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...
21
21 )()( iiii yyxx
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The length of the polygonal path is thus...
which is the approximate length of the curve
n
iiiii yyxx
1
21
21 )()(
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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...
iii xxx 1
iii yyy 1
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 don’t buy that…we can view x as a function of y when
we can’t view y as a function of x...
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To keep our discussion simple...
Assume that y is a function of xand that y is differentiablewith a continuous derivative
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Using the delta notation, we now have…
The length of the curve is approximately
n
iii yx
1
22 )()(
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Simplify the summands...
Factor out
))/(1()( 22iii xyx
And from And from therethere
2)/(1 iii xyx
2)( ix inside the radical to get
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Now the approximate arc length looks like...
i
n
iii xxy
1
2)/(1
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To get the actual arc length L?Let 0 ix
That gives usThat gives us
dxdxdyLb
a 2)/(1
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What? Where’d you get that?
)()(10
lim
b
a
n
iii
xdxxFxxF
i
Where the limit is taken over all partitions Where the limit is taken over all partitions
baxx n , interval theof ,...,0
AndAnd 1 iii xxx
Recall that
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In this setting...Playing the role of F(xi) we have
2)/(1 ii xy
And to make things more interestingAnd to make things more interesting than usual,than usual,
dxdyxy iixi
//lim0
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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 y=f(x) on the interval [a,b]
then the length L of the curve is given by
dxxfLb
a
))('(1 2
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