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Applications of Integration: Arc Length

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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  1. Applications of Integration: Arc Length Dr. Dillon Calculus II Fall 1999 SPSU

  2. Start with something easy The length of the line segment joining points (x0,y0) and (x1,y1) is (x1,y1) (x0,y0)

  3. The Length of a Polygonal Path? Add the lengths of the line segments.

  4. The length of a curve? Approximate by chopping it into polygonal pieces and adding up the lengths of the pieces

  5. Approximate the curve with polygonal pieces?

  6. 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)

  7. The length of that ith segment is...

  8. The length of the polygonal path is thus... which is the approximate length of the curve

  9. 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

  10. For Ease of Calculation... Let and

  11. 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...

  12. To keep our discussion simple... Assume that y is a function of x and that y is differentiable with a continuous derivative

  13. Using the delta notation, we now have… The length of the curve is approximately

  14. Simplify the summands... Factor out inside the radical to get And from there

  15. Now the approximate arc length looks like...

  16. To get the actual arc length L? Let That gives us

  17. What? Where’d you get that? Recall that Where the limit is taken over all partitions And

  18. In this setting... Playing the role of F(xi) we have And to make things more interesting than usual,

  19. What are a and b? The x coordinates of the endpoints of the arc

  20. 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

  21. 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

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