1. Volumes By Cylindrical Shells
slice || rotation 

2. Volumes By Cylindrical Shells
slice || rotation 
y  f x
y
x

3. Volumes By Cylindrical Shells
slice || rotation 
y  f x
y
a b x

4. Volumes By Cylindrical Shells
slice || rotation 
y  f x
y
a x b x

5. Volumes By Cylindrical Shells
slice || rotation 
y  f x
y
a x b x

6. Volumes By Cylindrical Shells
slice || rotation 
y  f x
y
a x b x

7. Volumes By Cylindrical Shells
slice || rotation 
y  f x
y
a x b x
x

8. Volumes By Cylindrical Shells
slice || rotation 
y  f x
y
y
a x b x
x

9. Volumes By Cylindrical Shells
slice || rotation 
y  f x
y
y  f x
a x b x
x

10. Volumes By Cylindrical Shells
slice || rotation 
y  f x
y
y  f x
a x b x
2x x

11. Volumes By Cylindrical Shells
slice || rotation 
y  f x
y
y  f x
a x b x
2x x
A x   2x  f  x 

12. Volumes By Cylindrical Shells
slice || rotation 
y  f x
y
y  f x
a x b x
2x x
A x   2x  f  x 
V  2x  f  x   x

13. Volumes By Cylindrical Shells
slice || rotation 
y  f x
y
y  f x
a x b x
2x x
A x   2x  f  x 
V  2x  f  x   x
b
V  lim  2x  f  x   x
x 0
xa

14. Volumes By Cylindrical Shells
slice || rotation 
y  f x
y
y  f x
a x b x
2x x
A x   2x  f  x 
V  2x  f  x   x
b
V  lim  2x  f  x   x
x 0
xa
b
 2  xf  x dx
a

15. e.g. i  Find the volume generated when the area between y  4  x 2 and
the x axis is rotated about the x axis

16. e.g. i  Find the volume generated when the area between y  4  x 2 and
the x axis is rotated about the x axis
y
4
y  4  x2
-2 2 x

17. e.g. i  Find the volume generated when the area between y  4  x 2 and
the x axis is rotated about the x axis
y
4
y  4  x2
y
-2 2 x

18. e.g. i  Find the volume generated when the area between y  4  x 2 and
the x axis is rotated about the x axis
y
4
y  4  x2
y
-2 2 x

19. e.g. i  Find the volume generated when the area between y  4  x 2 and
the x axis is rotated about the x axis
y
4
y  4  x2
y
-2 2 x
y

20. e.g. i  Find the volume generated when the area between y  4  x 2 and
the x axis is rotated about the x axis
y
4
y  4  x2 1
y  x  4  y  2
-2 2 x
y
2x

21. e.g. i  Find the volume generated when the area between y  4  x 2 and
the x axis is rotated about the x axis
y
4
y  4  x2 1
y  x  4  y  2
-2 2 x
y
2x

22. e.g. i  Find the volume generated when the area between y  4  x 2 and
the x axis is rotated about the x axis
y
4
y  4  x2 1
y  x  4  y  2
-2 2 x
y
1
2 x  24  y  2

23. e.g. i  Find the volume generated when the area between y  4  x 2 and
the x axis is rotated about the x axis
y
4
y  4  x2 1
y  x  4  y  2
-2 2 x
y
2y
1
2 x  24  y  2

24. e.g. i  Find the volume generated when the area between y  4  x 2 and
the x axis is rotated about the x axis
y
4
y  4  x2 1
y  x  4  y  2
-2 2 x
y
A y   2y 24  y 1 
 
2
 
2y
1
2 x  24  y  2

25. e.g. i  Find the volume generated when the area between y  4  x 2 and
the x axis is rotated about the x axis
y
4
y  4  x2 1
y  x  4  y  2
-2 2 x
y
A y   2y 24  y 1 
 
2
 
1
2y V  4y 4  y 2  y
1
2 x  24  y  2

26. e.g. i  Find the volume generated when the area between y  4  x 2 and
the x axis is rotated about the x axis 4 1
V  lim  4y 4  y 2  y
y y 0
x 0
4
y  4  x2 1
y  x  4  y  2
-2 2 x
y
A y   2y 24  y 1 
 
2
 
1
2y V  4y 4  y 2  y
1
2 x  24  y  2

27. e.g. i  Find the volume generated when the area between y  4  x 2 and
the x axis is rotated about the x axis 4 1
V  lim  4y 4  y 2  y
y y 0
x 0
4
y  4 x 2 4 1
y
1
 x  4  y  2  4  y 4  y 2 dy
0
-2 2 x
y
A y   2y 24  y 1 
 
2
 
1
2y V  4y 4  y 2  y
1
2 x  24  y  2

28. e.g. i  Find the volume generated when the area between y  4  x 2 and
the x axis is rotated about the x axis 4 1
V  lim  4y 4  y 2  y
y y 0
x 0
4
y  4 x 2 4 1
y
1
 x  4  y  2  4  y 4  y 2 dy
0
4 1
-2 2 x  4  4  y  y dy
2
0
y
A y   2y 24  y 1 
 
2
 
1
2y V  4y 4  y 2  y
1
2 x  24  y  2

29. e.g. i  Find the volume generated when the area between y  4  x 2 and
the x axis is rotated about the x axis 4 1
V  lim  4y 4  y 2  y
y y 0
x 0
4
y  4 x 2 4 1
y
1
 x  4  y  2  4  y 4  y 2 dy
0
4 1
-2 2 x  4  4  y  y dy
2
0
y
4
 1 3

 4   4 y  y dy

2 2
24  y 1  0 
A y   2y
 
2
 
1
2y V  4y 4  y 2  y
1
2 x  24  y  2

30. e.g. i  Find the volume generated when the area between y  4  x 2 and
the x axis is rotated about the x axis 4 1
V  lim  4y 4  y 2  y
y y 0
x 0
4
y  4 x 2 4 1
y
1
 x  4  y  2  4  y 4  y 2 dy
0
4 1
-2 2 x  4  4  y  y dy
2
0
y
4
 1 3

 4   4 y  y dy

2 2
24  y 1  0 
A y   2y 5 4
 
2
  8 2 2 2 
3
1  4  y  y 
2y V  4y 4  y 2  y 3 5 0
1
2 x  24  y  2

31. e.g. i  Find the volume generated when the area between y  4  x 2 and
the x axis is rotated about the x axis 4 1
V  lim  4y 4  y 2  y
y y 0
x 0
4
y  4 x 2 4 1
y
1
 x  4  y  2  4  y 4  y 2 dy
0
4 1
-2 2 x  4  4  y  y dy
2
0
y
4
 1 3

 4   4 y  y dy

2 2
24  y 1  0 
A y   2y 5 4
 
2
  8 2 2 2 
3
1  4  y  y 
2y V  4y 4  y 2  y 3 5 0
8    2 32   0
 4  8 
 3 5 
512

1
2 x  24  y  2
15
units 3

32. ii  Find the volume generated when x 2  y 2  4 is rotated
about the line x  4

33. ii  Find the volume generated when x 2  y 2  4 is rotated
about the line x  4
y
2
x2  y2  4
-2 2 x
-2

34. ii  Find the volume generated when x 2  y 2  4 is rotated
about the line x  4
y
2
x2  y2  4
-2 2 4 x
-2

35. ii  Find the volume generated when x 2  y 2  4 is rotated
about the line x  4
y
2
x2  y2  4
-2 2 4 x
-2 x

36. ii  Find the volume generated when x 2  y 2  4 is rotated
about the line x  4
y
2
x2  y2  4
-2 2 4 x
-2 x

37. ii  Find the volume generated when x 2  y 2  4 is rotated
about the line x  4
y
2
x2  y2  4
-2 2 4 x
-2 x

38. ii  Find the volume generated when x 2  y 2  4 is rotated
about the line x  4
y torus
2
x2  y2  4
-2 2 4 x
-2 x

39. ii  Find the volume generated when x 2  y 2  4 is rotated
about the line x  4
y torus
2
x2  y2  4
-2 2 4 x
-2 x

40. ii  Find the volume generated when x 2  y 2  4 is rotated
about the line x  4
y torus
2
x2  y2  4
-2 2 4 x
-2 x
x

41. ii  Find the volume generated when x 2  y 2  4 is rotated
about the line x  4
y torus
2
x2  y2  4
-2 2 4 x
-2 x
2y
x

42. ii  Find the volume generated when x 2  y 2  4 is rotated
about the line x  4
y torus
2
x2  y2  4
-2 2 4 x
-2 x
2y
 2 4  x2
x

43. ii  Find the volume generated when x 2  y 2  4 is rotated
about the line x  4
y torus
2
x2  y2  4
-2 2 4 x
-2 x
2y
 2 4  x2
2 4  x  x

44. ii  Find the volume generated when x 2  y 2  4 is rotated
about the line x  4
y torus
2
x2  y2  4
-2 2 4 x
-2 x
A x   2 4  x 2  2 4  x 
2y
 2 4  x2
2 4  x  x

45. ii  Find the volume generated when x 2  y 2  4 is rotated
about the line x  4
y torus
2
x2  y2  4
-2 2 4 x
-2 x
A x   2 4  x 2  2 4  x 
2y  4 4  x  4  x 2
 2 4  x2
2 4  x  x

46. ii  Find the volume generated when x 2  y 2  4 is rotated
about the line x  4
y torus
2
x2  y2  4
-2 2 4 x
-2 x
A x   2 4  x 2  2 4  x 
2y  4 4  x  4  x 2
 2 4  x2
V  4 4  x  4  x 2  x
2 4  x  x

47. 2
V  lim
x  0
 4 4  x  4  x 2  x
x  2

48. 2
V  lim
x  0
 4 4  x  4  x 2  x
x  2
2
 4  4  x  4  x 2 dx
2

49. 2
V  lim
x  0
 4 4  x  4  x 2  x
x  2
2
 4  4  x  4  x 2 dx
2
2 2
 4  4 4  x 2 dx  4  x 4  x 2 dx
2 2

50. 2
V  lim
x  0
 4 4  x  4  x 2  x
x  2
2
 4  4  x  4  x 2 dx
2
2 2
 4  4 4  x 2 dx  4  x 4  x 2 dx
2 2
2 2
 16  4  x 2 dx  4  x 4  x 2 dx
2 2

51. 2
V  lim
x  0
 4 4  x  4  x 2  x
x  2
2
 4  4  x  4  x 2 dx
2
2 2
 4  4 4  x 2 dx  4  x 4  x 2 dx
2 2
2 2
 16  4  x 2 dx  4  x 4  x 2 dx
2 2
semi-circle

52. 2
V  lim
x  0
 4 4  x  4  x 2  x
x  2
2
 4  4  x  4  x 2 dx
2
2 2
 4  4 4  x 2 dx  4  x 4  x 2 dx
2 2
2 2
 16  4  x 2 dx  4  x 4  x 2 dx
2 2
semi-circle
 16   2  
1 2
2
 


53. 2
V  lim
x  0
 4 4  x  4  x 2  x
x  2
2
 4  4  x  4  x 2 dx
2
2 2
 4  4 4  x 2 dx  4  x 4  x 2 dx
2 2
2 2
 16  4  x 2 dx  4  x 4  x 2 dx
2 2
semi-circle odd  even  odd function
 16   2  
1 2
2
 


54. 2
V  lim
x  0
 4 4  x  4  x 2  x
x  2
2
 4  4  x  4  x 2 dx
2
2 2
 4  4 4  x 2 dx  4  x 4  x 2 dx
2 2
2 2
 16  4  x 2 dx  4  x 4  x 2 dx
2 2
semi-circle odd  even  odd function
 16   2    0
1 2
2
 


55. 2
V  lim
x  0
 4 4  x  4  x 2  x
x  2
2
 4  4  x  4  x 2 dx
2
2 2
 4  4 4  x 2 dx  4  x 4  x 2 dx
2 2
2 2
 16  4  x 2 dx  4  x 4  x 2 dx
2 2
semi-circle odd  even  odd function
 16   2    0
1 2
2
 

 32 2 units 3

56. 2
V  lim
x  0
 4 4  x  4  x 2  x
x  2
2
 4  4  x  4  x 2 dx Exercise 3B;
2 3, 4, 6, 8
2 2
 4  4 4  x 2 dx  4  x 4  x 2 dx Exercise 3C;
2 2
2 2
1, 3, 9, 12, 20
 16  4  x 2 dx  4  x 4  x 2 dx
2 2
semi-circle odd  even  odd function
 16   2    0
1 2
2
 

 32 2 units 3