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)

Slide 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

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

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

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

Slide 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 

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

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

Slide 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

Slide 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

Slide 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].

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