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Applications of Integration: Arc Length - PowerPoint PPT Presentation

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

Dr. Dillon

Calculus II

Fall 1999

SPSU

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

(x1,y1)

(x0,y0)

Add the lengths of the line segments.

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

• 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...

which is the approximate 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

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...

Assume that y is a function of x

and that y is differentiable

with a continuous derivative

The length of the curve is approximately

Factor out

And from there

Let

That gives us

Recall that

Where the limit is taken over all partitions

And

Playing the role of F(xi) we have

And to make things more interesting

than usual,

The x coordinates of the endpoints of the arc

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

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

then the length L of the curve is given by