1. Volumes of Solids
Volumes By Discs & Washers
slice  rotation 

2. Volumes of Solids
Volumes By Discs & Washers
slice  rotation 
About the x axis

3. Volumes of Solids
Volumes By Discs & Washers
slice  rotation 
About the x axis
y  f x
y
x

4. Volumes of Solids
Volumes By Discs & Washers
slice  rotation 
About the x axis
y  f x
y
a b x

5. Volumes of Solids
Volumes By Discs & Washers
slice  rotation 
About the x axis
y  f x
y
a x b x

6. Volumes of Solids
Volumes By Discs & Washers
slice  rotation 
About the x axis
y  f x
y
y
a x b x
x

7. Volumes of Solids
Volumes By Discs & Washers
slice  rotation 
About the x axis
y  f x
y
y  f x
a x b x
x

8. Volumes of Solids
Volumes By Discs & Washers
slice  rotation 
About the x axis
y  f x
y
y  f x
A x     f  x 
2
a x b x
x

9. Volumes of Solids
Volumes By Discs & Washers
slice  rotation 
About the x axis
y  f x
y
y  f x
A x     f  x 
2
a x b x V    f  x   x
2
x

10. Volumes of Solids
Volumes By Discs & Washers
slice  rotation 
About the x axis
y  f x
y
y  f x
A x     f  x 
2
a x b x V    f  x   x
2
x
b
V  lim    f  x   x
2
x 0
xa

11. Volumes of Solids
Volumes By Discs & Washers
slice  rotation 
About the x axis
y  f x
y
y  f x
A x     f  x 
2
a x b x V    f  x   x
2
x
b
V  lim    f  x   x
2
x 0
xa
b
    f  x  dx
2
a

12. About the y axis
y  f x
y
x

13. About the y axis
y  f x
y
d
c
x

14. About the y axis
y  f x
y
d
y
c
x

15. About the y axis
y  f x
y
d
y x
y
c
x

16. About the y axis
y  f x
y
d
y x  g y
y
c
x

17. About the y axis
y  f x
y
d
y x  g y
y
c
A y    g  y 
2
x

18. About the y axis
y  f x
y
d
y x  g y
y
c
A y    g  y 
2
x
V   g  y   y
2

19. About the y axis
y  f x
y
d
y x  g y
y
c
A y    g  y 
2
x
V   g  y   y
2
d
V  lim   g  y   y
2
y  0
y c

20. About the y axis
y  f x
y
d
y x  g y
y
c
A y    g  y 
2
x
V   g  y   y
2
d
V  lim   g  y   y
2
y  0
y c
d
   g  y  dy
2
c

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

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
-2 2 x

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
-2 x 2 x

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
-2 x 2 x
y  4  x2
x

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
-2 x 2 x
y  4  x2
A x    4  x 
2 2
x

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
y
4
y  4  x2
-2 x 2 x
y  4  x2
A x    4  x 
2 2
V   16  8 x 2  x 4  x
x

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
V  lim   16  8 x 2  x 4  x
2
y
4 x 0
x  2
y  4 x 2
-2 x 2 x
y  4  x2
A x    4  x 
2 2
V   16  8 x 2  x 4  x
x

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
V  lim   16  8 x 2  x 4  x
2
y
4 x 0
x  2
y  4 x 2
2
 2  16  8 x 2  x 4 dx
-2 x 2 x
0
y  4  x2
A x    4  x 
2 2
V   16  8 x 2  x 4  x
x

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
V  lim   16  8 x 2  x 4  x
2
y
4 x 0
x  2
y  4 x 2
2
 2  16  8 x 2  x 4 dx
-2 x 2 x
0
2
16 x  8 x 3  1 x 5 
 2 
 3 5 0 
y  4  x2
A x    4  x 
2 2
V   16  8 x 2  x 4  x
x

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
V  lim   16  8 x 2  x 4  x
2
y
4 x 0
x  2
y  4 x 2
2
 2  16  8 x 2  x 4 dx
-2 x 2 x
0
2
16 x  8 x 3  1 x 5 
 2 
 3 5 0 
32 64 32 0
 2     
 3 5 
y  4  x2
512
 units 3
A x    4  x 
2 2 15
V   16  8 x 2  x 4  x
x

31. ii  Find the volume generated when the area enclosed by y  4  x 2 and
y  3 is rotated about the line y  3

32. ii  Find the volume generated when the area enclosed by y  4  x 2 and
y  3 is rotated about the line y  3
y
4
y3
x
y  4  x2

33. ii  Find the volume generated when the area enclosed by y  4  x 2 and
y  3 is rotated about the line y  3
y
4
(-1,3) (1,3) y3
x
y  4  x2

34. ii  Find the volume generated when the area enclosed by y  4  x 2 and
y  3 is rotated about the line y  3
y
4
(-1,3) (1,3) y3
x
x
y  4  x2

35. ii  Find the volume generated when the area enclosed by y  4  x 2 and
y  3 is rotated about the line y  3
y
4
(-1,3) (1,3) y3
x
x
y  4  x2
x

36. ii  Find the volume generated when the area enclosed by y  4  x 2 and
y  3 is rotated about the line y  3
y
4
(-1,3) (1,3) y3
x
x
y  4  x2
y  3  4  x2  3
 1 x2
x

37. ii  Find the volume generated when the area enclosed by y  4  x 2 and
y  3 is rotated about the line y  3
y
4
(-1,3) (1,3) y3
x
x
y  4  x2
y  3  4  x2  3
 1 x2
A x    1  x 
2 2
x

38. ii  Find the volume generated when the area enclosed by y  4  x 2 and
y  3 is rotated about the line y  3
y
4
(-1,3) (1,3) y3
x
x
y  4  x2
y  3  4  x2  3
 1 x2
A x    1  x 
2 2
V   1  2 x 2  x 4  x x

39. ii  Find the volume generated when the area enclosed by y  4  x 2 and
y  3 is rotated about the line y  3
 1  2 x 2  x 4  x
1
y
4 V  lim
x  0

x  1
(-1,3) (1,3) y3
x
x
y  4  x2
y  3  4  x2  3
 1 x2
A x    1  x 
2 2
V   1  2 x 2  x 4  x x

40. ii  Find the volume generated when the area enclosed by y  4  x 2 and
y  3 is rotated about the line y  3
 1  2 x 2  x 4  x
1
y
4 V  lim
x  0

x  1
(-1,3) (1,3) y3 1
x  2  1  2 x 2  x 4 dx
0
x
y  4  x2
y  3  4  x2  3
 1 x2
A x    1  x 
2 2
V   1  2 x 2  x 4  x x

41. ii  Find the volume generated when the area enclosed by y  4  x 2 and
y  3 is rotated about the line y  3
 1  2 x 2  x 4  x
1
y
4 V  lim
x  0

x  1
(-1,3) (1,3) y3 1
x  2  1  2 x 2  x 4 dx
0
x 1
 x  2 x3  1 x5 
 2 
y  4  x2
 3 5 0 
y  3  4  x2  3
 1 x2
A x    1  x 
2 2
V   1  2 x 2  x 4  x x

42. ii  Find the volume generated when the area enclosed by y  4  x 2 and
y  3 is rotated about the line y  3
 1  2 x 2  x 4  x
1
y
4 V  lim
x  0

x  1
(-1,3) (1,3) y3 1
x  2  1  2 x 2  x 4 dx
0
x 1
 x  2 x3  1 x5 
 2 
y  4  x2
 3 5 0 
y  3  4  x2  3 1 2 1 0
 2     
 3 5 
 1 x2
16
 units 3
A x    1  x 
2 2
15
V   1  2 x 2  x 4  x x

43. iii  The area between the curves y  x 2 and x  y 2 is rotated
about the line y axis. Find the volume generated.

44. iii  The area between the curves y  x 2 and x  y 2 is rotated
about the line y axis. Find the volume generated.
y y  x2
x  y2
x

45. iii  The area between the curves y  x 2 and x  y 2 is rotated
about the line y axis. Find the volume generated.
y y  x2
x  y2
(1,1)
x

46. iii  The area between the curves y  x 2 and x  y 2 is rotated
about the line y axis. Find the volume generated.
y y  x2
x  y2
(1,1)
y
x

47. iii  The area between the curves y  x 2 and x  y 2 is rotated
about the line y axis. Find the volume generated.
y y  x2
x  y2
(1,1)
y
x

48. iii  The area between the curves y  x 2 and x  y 2 is rotated
about the line y axis. Find the volume generated.
y y  x2
x  y2
(1,1)
y
x
1
x y 2

49. iii  The area between the curves y  x 2 and x  y 2 is rotated
about the line y axis. Find the volume generated.
y y  x2
x  y2
(1,1)
y
x
1
x y 2
x y 2

50. iii  The area between the curves y  x 2 and x  y 2 is rotated
about the line y axis. Find the volume generated.
y y  x2
x  y2
(1,1)
y
x
1
x y 2
x y 2
 1  2 
A y     y    y  

2 2 2
 
 


51. iii  The area between the curves y  x 2 and x  y 2 is rotated
about the line y axis. Find the volume generated.
y y  x2
x  y2
(1,1)
y
x
1
x y 2
x y 2
 1  2 
A y     y    y  

2 2 2
 
 

V    y  y 4  y

52. iii  The area between the curves y  x 2 and x  y 2 is rotated
about the line y axis. Find the volume generated.
y y  x2
V  lim    y  y 4  y
1
x y 2
y 0
y 0
(1,1)
y
x
1
x y 2
x y 2
 1  2 
A y     y    y  

2 2 2
 
 

V    y  y 4  y

53. iii  The area between the curves y  x 2 and x  y 2 is rotated
about the line y axis. Find the volume generated.
y y  x2
V  lim    y  y 4  y
1
x y 2
y 0
y 0
(1,1) 1
y     y  y 4 dy
0
x
1
x y 2
x y 2
 1  2 
A y     y    y  

2 2 2
 
 

V    y  y 4  y

54. iii  The area between the curves y  x 2 and x  y 2 is rotated
about the line y axis. Find the volume generated.
y y  x2
V  lim    y  y 4  y
1
x y 2
y 0
y 0
(1,1) 1
y     y  y 4 dy
0
x
1
1 2 1 5 
 y  y 
1 2 5 0
x y 2
x y 2
 1  2 
A y     y    y  

2 2 2
 
 

V    y  y 4  y

55. iii  The area between the curves y  x 2 and x  y 2 is rotated
about the line y axis. Find the volume generated.
y y  x2
V  lim    y  y 4  y
1
x y 2
y 0
y 0
(1,1) 1
y     y  y 4 dy
0
x
1
1 2 1 5 
 y  y 
1 2 5 0
x y 2
x y 2
 1 1 0
   
2 5 
3
 1  2   units 3
A y     y    y  

2 2 2 10
 
 

V    y  y 4  y

56. (iv) (1996) y
1
x  y2  0 S
1 4 x
-1
The shaded region is bounded by the lines x = 1, y = 1 and y = -1 and
the curve x  y  0 . The region is rotated through 360 about the line
2
x = 4 to form a solid.
When the region is rotated, the line segment S at height y sweeps out
an annulus.

57. a) Show that the area of the annulus at height y is equal to   y 4  8 y 2  7 

58. a) Show that the area of the annulus at height y is equal to   y 4  8 y 2  7 
y

59. a) Show that the area of the annulus at height y is equal to   y 4  8 y 2  7 
y r

60. a) Show that the area of the annulus at height y is equal to   y 4  8 y 2  7 
R
y

61. 1
x  y2  0 S
1 4 x
-1

62. 1
x  y2  0 S r
1 4 x
-1

63. 1
x  y2  0 S r  4 1
3
1 4 x
-1

64. 1
x  y2  0 S r  4 1
3
1 4 x
R
-1

65. 1
x  y2  0 S r  4 1
3
1 4 x
R  4 x
-1  4  y2

66. a) Show that the area of the annulus at height y is equal to   y 4  8 y 2  7 
y

67. a) Show that the area of the annulus at height y is equal to   y 4  8 y 2  7 
R  4 x
 4  y2
y

68. a) Show that the area of the annulus at height y is equal to   y 4  8 y 2  7 
R  4 x
 4  y2
y
r  4 1
3

69. a) Show that the area of the annulus at height y is equal to   y 4  8 y 2  7 
R  4 x
 4  y2
y
r  4 1

A y    4  y   3 
2 2 2
3

70. a) Show that the area of the annulus at height y is equal to   y 4  8 y 2  7 
R  4 x
 4  y2
y
r  4 1

A y    4  y   3 
2 2 2
3
  16  8 y 2  y 4  9 
  y 4  8 y 2  7

71. b) Hence find the volume of the solid.

72. b) Hence find the volume of the solid.
V    y 4  8 y 2  7  y

73. b) Hence find the volume of the solid.
V    y 4  8 y 2  7  y
  y 4  8 y 2  7  y
1
V  lim
y  0

y  1

74. b) Hence find the volume of the solid.
V    y 4  8 y 2  7  y
  y 4  8 y 2  7  y
1
V  lim
y  0

y  1
1
 2   y 4  8 y 2  7 dy
0

75. b) Hence find the volume of the solid.
V    y 4  8 y 2  7  y
  y 4  8 y 2  7  y
1
V  lim
y  0

y  1
1
 2   y 4  8 y 2  7 dy
0
1
 2  y 5  y 3  7 y 
1 8
5
 3 0


76. b) Hence find the volume of the solid.
V    y 4  8 y 2  7  y
  y 4  8 y 2  7  y
1
V  lim
y  0

y  1
1
 2   y 4  8 y 2  7 dy
0
1
 2  y 5  y 3  7 y 
1 8
5
 3 0

1  8   
 2  7 0
5 3 
296
 units 3
15

77. b) Hence find the volume of the solid.
V    y 4  8 y 2  7  y
  y 4  8 y 2  7  y
1
V  lim
y  0

y  1
1
 2   y 4  8 y 2  7 dy Exercise 3A;
0
1 2, 5, 7, 9, 13, 14, 15
 2  y 5  y 3  7 y 
1 8
5
 3 0
 Exercise 3B;
1  8   
 2  1, 2, 5, 7, 9, 10, 11, 12
7 0
5 3 
296
 units 3
15