Applications of Integration: Arc Length

Dr. DillonCalculus IIFall 1999

SPSU

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)

The Length of a Polygonal Path?

Add the lengths of the line segments.

The length of a curve?

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

Approximate the curve with polygonal pieces?

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 ith segment is...

21

21 )()( iiii yyxx

The length of the polygonal path is thus...

which is the approximate length of the curve

n

iiiii yyxx

1

21

21 )()(

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

For Ease of Calculation...

iii xxx 1

iii yyy 1

Let

and

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...

Assume that y is a function of xand that y is differentiablewith a continuous derivative

Using the delta notation, we now have…

The length of the curve is approximately

n

iii yx

1

22 )()(

Simplify the summands...

Factor out

))/(1()( 22iii xyx

And from And from therethere

2)/(1 iii xyx

2)( ix inside the radical to get

Now the approximate arc length looks like...

i

n

iii xxy

1

2)/(1

To get the actual arc length L?Let 0 ix

That gives usThat gives us

dxdxdyLb

a 2)/(1

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

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

What are a and b?

The x coordinates of the endpoints of the arc

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?

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