1. Applications of Integration Volumes of Revolution Many thanks to http://mathdemos.gcsu.edu/shellmeth od/gallery/gallery.html

2. Method of discs

3. Take this ordinary line 2 5 Revolve this line around the x axis We form a cylinder of volume

4. We could find the volume by finding the volume of small disc sections 2 5

5. If we stack all these slices… We can sum all the volumes to get the total volume

6. To find the volume of a cucumber… we could slice the cucumber into discs and find the volume of each disc.

7. The volume of one section: Volume of one slice =

8. We could model the cucumber with a mathematical curve and revolve this curve around the x axis… Each slice would have a thickness dx and height y. 25 -5

9. The volume of one section: r = y value h = dx Volume of one slice =

10. Volume of cucumber… Area of 1 slice Thickness of slice

11. Take this function… and revolve it around the x axis

12. We can slice it up, find the volume of each disc and sum the discs to find the volume….. Radius = y Area = Thickness of slice = dx Volume of one slice=

13. Take this shape…

14. Revolve it…

15. Christmas bell…

16. Divide the region into strips

17. Form a cylindrical slice

18. Repeat the procedure for each strip

19. To generate this solid

27. A polynomial

30. Regions that can be revolved using disc method

31. Regions that cannot….

32. Model this muffin.

33. Washer Method

34. A different cake

35. Slicing….

36. Making a washer

38. Revolving around the x axis

39. Region bounded between y = 1, x = 0, y = 1 x = 0

40. Volume generated between two curves y= 1

41. Area of cross section.. f(x) g(x)

42. dx

43. Your turn: Region bounded between x = 0, y = x,

44. Region bounded between y =1, x = 1

48. Region bounded between

49. Around the x axis- set it up

50. Revolving shapes around the y axis

51. Region bounded between

53. Volume of one washer is

54. Calculate the volume of one washer

55. And again…region bounded between y=sin(x), y = 0.

57. Region bounded between x = 0, y = 0, x = 1,

59. Worksheet 5 Delta Exercise 16.5