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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

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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)

### 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

### 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 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?

• 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

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 x

and that y is differentiable

with a continuous derivative

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

The length of the curve is approximately

### Simplify the summands...

Factor out

inside the radical to get

And from there

Let

That gives us

### What? Where’d you get that?

Recall that

Where the limit is taken over all partitions

And

### In this setting...

Playing the role of F(xi) we have

And to make things more interesting

than usual,

### 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