2. 1. 2. Warm-Up:

3. 1. Warm-Up:

4. 2. Warm-Up:

5. HW ?

6. 7-2: Volume of Solids of Revolution Find volume using the disk and washer methods Find volume of solids with known cross sections ©2002Roy L. Gover (www.mrgover.com) Objectives:

7. Examples Created by Lawrence Hunsh

8. Example from [0,2] Rotate about x axis Creates Partition with disks

9. Important Idea x y Each cylinder is a disk Solid of Revolution

10. Important Idea Volume of ea. disk=  r i 2  x riri xx R

11. The total volume is given by the Riemann Sum: The definite integral is the accumulator of the disk volumes

12. The Disk Method To find the volume of a solid of revolution, use one of the following: Hori. Axis of Rev. Vert. Axis of Rev.

13. Example Find the volume of the solid of revolution formed by revolving the graph: around the x axis. R is a constant

14. Assignment 1. 463/1-4 all, Slides 1-15 Slides 16-23 2. 464/9-12 all 37,39,43 3. 463/7,8,13,17,19-23 odd, 51,52

15. Warm-Up Find the volume of the solid of revolution formed by revolving the graph around thefrom x=0 to x=1 x axis.Show your integral setup and evaluate with your calculator. Your answer should be accurate to 3 decimal places.

16. Solution 0 1Setup Top half of Disks

17. Try This Find the volume of the solid of revolution formed by revolving the graph around the x axis. 0  cu. units

18. Example Find the volume of the solid of revolution formed by revolving the graph around the y axis. 0 1

19. Try This Find the volume of the solid of revolution formed by revolving the 0 2 region bounded by y= 2 x 2, x= 0 & x= 2 about the y axis.

20. Example Find the volume of the solid formed by revolving the region bounded by the graphs y= 2 x 2, y= 0 & x= 2 about the line x= 2. 8 02

21. Try This Find the volume of the solid formed by revolving the region bounded by the graphs y=x 2, y= 0 & x= 3 about the line x= 3. 0 3

22. The Washer Method If the disk has a hole, it is a washer R r Vol. Of washer

23. Important Idea The volume of a solid of revolution with a hole is the volume of the solid without the hole less the volume of the hole.

24. Example Find the volume of the solid formed by revolving the region bounded by the graphs y=x and y=x 2 about the x axis. R r

25. Example Find the volume of the solid formed by revolving the region bounded by the graphs y=x and y=x 2 about the y axis. R r

26. Assignment 1. 463/1-4all Slides 16-23 2. 463/7,8,13,17, 37,39,43

27. Warm-Up Find the volume of the solid generated when the region between the graphs f(x)= 1/2 over the interval [0,2] is revolved about the x axis. + x 2 and g(x)=x

28. Example Find the volume of the solid formed by revolving the region bounded by the graphs y= 2 x 2, y= 0 & x = 2 about the line y= 8. 8 0 2

29. Try This Find the volume of the solid formed by revolving the region bounded by the graphs, y= 0 & x = 1, x =4, about the line y= 4. 1 4 0 4

30. Solution 1 4 0 4

31. Solids with Known Cross Sections x y

32. Example Find the volume of the solid whose base is bounded by the circle x 2 +y 2 =4 with equilateral triangle cross sections perpendicular to the x axis. x y

33. Try This Find the volume of the solid whose base is bounded by the circle x 2 +y 2 =4 with semicircular cross sections perpendicular to the x axis. x y

34. Lesson Close Write a paragraph describing how you find the volume of a solid of revolution. Dr. Lou Talman, Metro State University, Denver, CO. "Solids of Revolution." [Online image] 29 December 2004.. Unknown Author,"Solids of Revolution." [Online image] December 2004.. Lawrence S. Husch, University of Tenn., Knoxville. "Visual Calculus-Solids of Revolution." [Online image] 29 December 2004.. Credits-animated pictures on slides 1,2 and 3

35. Assignment 1. 463/1-6 all, 37,39,43 Slides 1-15 Slides 16-23 2. 464/9-12 all 3. 463/7,8,13,17,19-23 odd, 51,52 (Disks about y Axis)