Loading in 5 sec....

Applications of Integration: Arc LengthPowerPoint Presentation

Applications of Integration: Arc Length

- 74 Views
- Uploaded on
- Presentation posted in: General

Applications of Integration: Arc Length

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

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)

which is the approximate 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

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

inside the radical to get

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