1. Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.3 Lengths of Plane Curves (1 st lecture of week 10/09/07- 15/09/07)

2. Slide 6 - 2 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Length of a parametrically defined curve L k the line segment between P k and P k-1

3. Slide 6 - 3 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

4. Slide 6 - 4 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

5. Slide 6 - 5 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

6. Slide 6 - 6 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1 The circumference of a circle  Find the length of the circle of radius r defined parametrically by  x=r cos t and y=r sin t, 0 ≤ t ≤ 2 

7. Slide 6 - 7 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Length of a curve y = f(x)

8. Slide 6 - 8 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

9. Slide 6 - 9 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3 Applying the arc length formula for a graph  Find the length of the curve

10. Slide 6 - 10 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Dealing with discontinuity in dy/dx  At a point on a curve where dy/dx fails to exist and we may be able to find the curve’s length by expressing x as a function of y and applying the following

11. Slide 6 - 11 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 4 Length of a graph which has a discontinuity in dy/dx  Find the length of the curve y = (x/2) 2/3 from x = 0 to x = 2.  Solution  dy/dx = (1/3) (2/x) 1/3 is not defined at x=0.  dx/dy = 3y 1/2 is continuous on [0,1].

12. Slide 6 - 12 Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley