1. Volumes By Cylindrical Shells
( slice || rotation )

2. Volumes By Cylindrical Shells
( slice || rotation )
y y = f ( x)
x

3. Volumes By Cylindrical Shells
( slice || rotation )
y y = f ( x)
a b x

4. Volumes By Cylindrical Shells
( slice || rotation )
y y = f ( x)
a ∆x b x

5. Volumes By Cylindrical Shells
( slice || rotation )
y y = f ( x)
a ∆x b x

6. Volumes By Cylindrical Shells
( slice || rotation )
y y = f ( x)
a ∆x b x

7. Volumes By Cylindrical Shells
( slice || rotation )
y y = f ( x)
a ∆x b x
∆x

8. Volumes By Cylindrical Shells
( slice || rotation )
y y = f ( x)
y
a ∆x b x
∆x

9. Volumes By Cylindrical Shells
( slice || rotation )
y y = f ( x)
y = f ( x)
a ∆x b x
∆x

10. Volumes By Cylindrical Shells
( slice || rotation )
y y = f ( x)
y = f ( x)
a ∆x b x
∆x
2πx

11. Volumes By Cylindrical Shells
( slice || rotation )
y y = f ( x)
y = f ( x)
a ∆x b x
∆x
2πx
A( x ) = 2πx ⋅ f ( x )

12. Volumes By Cylindrical Shells
( slice || rotation )
y y = f ( x)
y = f ( x)
a ∆x b x
∆x
2πx
A( x ) = 2πx ⋅ f ( x )
∆V = 2πx ⋅ f ( x ) ⋅ ∆x

13. Volumes By Cylindrical Shells
( slice || rotation )
y y = f ( x)
y = f ( x)
a ∆x b x
∆x
2πx
A( x ) = 2πx ⋅ f ( x )
∆V = 2πx ⋅ f ( x ) ⋅ ∆x
b
V = lim ∑ 2πx ⋅ f ( x ) ⋅ ∆x
∆x →0
x=a

14. Volumes By Cylindrical Shells
( slice || rotation )
y y = f ( x)
y = f ( x)
a ∆x b x
∆x
2πx
A( x ) = 2πx ⋅ f ( x )
∆V = 2πx ⋅ f ( x ) ⋅ ∆x
b
V = lim ∑ 2πx ⋅ f ( x ) ⋅ ∆x
∆x →0
x=a
b
= 2π ∫ 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
x
-2 2

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
⇒ x = ( 4 − y) 2
∆y
-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
⇒ x = ( 4 − y) 2
∆y
-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
⇒ x = ( 4 − y) 2
∆y
-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
⇒ x = ( 4 − y) 2
∆y
-2 2 x
∆y
2πy
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
⇒ x = ( 4 − y) 2
∆y
-2 2 x
∆y
2( 4 − y ) 1 
A( y ) = 2πy 2
 
 
2πy
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
⇒ x = ( 4 − y) 2
∆y
-2 2 x
∆y
2( 4 − y ) 1 
A( y ) = 2πy 2
 
 
1
2πy ∆V = 4πy ( 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 ∑ 4πy ( 4 − y ) ⋅ ∆y
2
y ∆y →0
x =0
4
y = 4 − x2 1
⇒ x = ( 4 − y) 2
∆y
-2 2 x
∆y
2( 4 − y ) 1 
A( y ) = 2πy 2
 
 
1
2πy ∆V = 4πy ( 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 ∑ 4πy ( 4 − y ) ⋅ ∆y
2
y ∆y →0
x =0
4 4
y = 4 − x2 1
= 4π ∫ y ( 4 − y ) dy
1
⇒ x = ( 4 − y) 2
∆y 2
0
-2 2 x
∆y
2( 4 − y ) 1 
A( y ) = 2πy 2
 
 
1
2πy ∆V = 4πy ( 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 ∑ 4πy ( 4 − y ) ⋅ ∆y
2
y ∆y →0
x =0
4 4
y = 4 − x2 1
= 4π ∫ y ( 4 − y ) dy
1
⇒ x = ( 4 − y) 2
∆y 2
0
4 1
= 4π ∫ ( 4 − y ) y dy
2
-2 2 x
0
∆y
2( 4 − y ) 1 
A( y ) = 2πy 2
 
 
1
2πy ∆V = 4πy ( 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 ∑ 4πy ( 4 − y ) ⋅ ∆y
2
y ∆y →0
x =0
4 4
y = 4 − x2 1
= 4π ∫ y ( 4 − y ) dy
1
⇒ x = ( 4 − y) 2
∆y 2
0
4 1
= 4π ∫ ( 4 − y ) y dy
2
-2 2 x
0
1 
4 3
∆y = 4π ∫  4 y − y dy
2
2

0 
2( 4 − y ) 1 
A( y ) = 2πy 2
 
 
1
2πy ∆V = 4πy ( 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 ∑ 4πy ( 4 − y ) ⋅ ∆y
2
y ∆y →0
x =0
4 4
y = 4 − x2 1
= 4π ∫ y ( 4 − y ) dy
1
⇒ x = ( 4 − y) 2
∆y 2
0
4 1
= 4π ∫ ( 4 − y ) y dy
2
-2 2 x
0
1 
4 3
∆y = 4π ∫  4 y − y dy
2
2

0 4
2( 4 − y ) 1 
A( y ) = 2πy 2
  8 3 2 5 
  = 4π  y 2 − y 2 
1
2πy ∆V = 4πy ( 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 ∑ 4πy ( 4 − y ) ⋅ ∆y
2
y ∆y →0
x =0
4 4
y = 4 − x2 1
= 4π ∫ y ( 4 − y ) dy
1
⇒ x = ( 4 − y) 2
∆y 2
0
4 1
= 4π ∫ ( 4 − y ) y dy
2
-2 2 x
0
1 
4 3
∆y = 4π ∫  4 y − y dy
2
2

0 4
2( 4 − y ) 1 
A( y ) = 2πy 2
  8 3 2 5 
  = 4π  y 2 − y 2 
1
2πy ∆V = 4πy ( 4 − y ) 2 ⋅ ∆y 3 5 0
8 ( 8) − 2 ( 32 ) − 0
= 4π  
 
3 5
512π
1
= units 3
2 x = 2( 4 − y ) 2
15

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
∆x
2π ( 4 − 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 ⋅ 2π ( 4 − x )
2y
= 2 4 − x2
∆x
2π ( 4 − 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 ⋅ 2π ( 4 − x )
= 4π ( 4 − x ) 4 − x 2
2y
= 2 4 − x2
∆x
2π ( 4 − 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 ⋅ 2π ( 4 − x )
= 4π ( 4 − x ) 4 − x 2
2y
= 2 4 − x2
∆V = 4π ( 4 − x ) 4 − x 2 ⋅ ∆x
∆x
2π ( 4 − x )

47. 2
∑ 4π ( 4 − x ) 4 − x 2 ⋅ ∆x
V = lim
∆x →0
x = −2

48. 2
∑ 4π ( 4 − x ) 4 − x 2 ⋅ ∆x
V = lim
∆x →0
x = −2
2
= 4π ∫ ( 4 − x ) 4 − x 2 dx
−2

49. 2
∑ 4π ( 4 − x ) 4 − x 2 ⋅ ∆x
V = lim
∆x →0
x = −2
2
= 4π ∫ ( 4 − x ) 4 − x 2 dx
−2
2 2
= 4π ∫ 4 4 − x 2 dx − 4π ∫ x 4 − x 2 dx
−2 −2

50. 2
∑ 4π ( 4 − x ) 4 − x 2 ⋅ ∆x
V = lim
∆x →0
x = −2
2
= 4π ∫ ( 4 − x ) 4 − x 2 dx
−2
2 2
= 4π ∫ 4 4 − x 2 dx − 4π ∫ x 4 − x 2 dx
−2 −2
2 2
= 16π ∫ 4 − x 2 dx − 4π ∫ x 4 − x 2 dx
−2 −2

51. 2
∑ 4π ( 4 − x ) 4 − x 2 ⋅ ∆x
V = lim
∆x →0
x = −2
2
= 4π ∫ ( 4 − x ) 4 − x 2 dx
−2
2 2
= 4π ∫ 4 4 − x 2 dx − 4π ∫ x 4 − x 2 dx
−2 −2
2 2
= 16π ∫ 4 − x 2 dx − 4π ∫ x 4 − x 2 dx
−2 −2
semi-circle

52. 2
∑ 4π ( 4 − x ) 4 − x 2 ⋅ ∆x
V = lim
∆x →0
x = −2
2
= 4π ∫ ( 4 − x ) 4 − x 2 dx
−2
2 2
= 4π ∫ 4 4 − x 2 dx − 4π ∫ x 4 − x 2 dx
−2 −2
2 2
= 16π ∫ 4 − x 2 dx − 4π ∫ x 4 − x 2 dx
−2 −2
semi-circle
= 16π  π ( 2) 
1 2
2 
 

53. 2
∑ 4π ( 4 − x ) 4 − x 2 ⋅ ∆x
V = lim
∆x →0
x = −2
2
= 4π ∫ ( 4 − x ) 4 − x 2 dx
−2
2 2
= 4π ∫ 4 4 − x 2 dx − 4π ∫ x 4 − x 2 dx
−2 −2
2 2
= 16π ∫ 4 − x 2 dx − 4π ∫ x 4 − x 2 dx
−2 −2
odd × even = odd function
semi-circle
= 16π  π ( 2) 
1 2
2 
 

54. 2
∑ 4π ( 4 − x ) 4 − x 2 ⋅ ∆x
V = lim
∆x →0
x = −2
2
= 4π ∫ ( 4 − x ) 4 − x 2 dx
−2
2 2
= 4π ∫ 4 4 − x 2 dx − 4π ∫ x 4 − x 2 dx
−2 −2
2 2
= 16π ∫ 4 − x 2 dx − 4π ∫ x 4 − x 2 dx
−2 −2
odd × even = odd function
semi-circle
= 16π  π ( 2)  − 0
1 2
2 
 

55. 2
∑ 4π ( 4 − x ) 4 − x 2 ⋅ ∆x
V = lim
∆x →0
x = −2
2
= 4π ∫ ( 4 − x ) 4 − x 2 dx
−2
2 2
= 4π ∫ 4 4 − x 2 dx − 4π ∫ x 4 − x 2 dx
−2 −2
2 2
= 16π ∫ 4 − x 2 dx − 4π ∫ x 4 − x 2 dx
−2 −2
odd × even = odd function
semi-circle
= 16π  π ( 2)  − 0
1 2
2 
 
= 32π 2 units 3

56. 2
∑ 4π ( 4 − x ) 4 − x 2 ⋅ ∆x
V = lim
∆x →0
x = −2
2
= 4π ∫ ( 4 − x ) 4 − x 2 dx Exercise 3B;
3, 4, 6, 8
−2
2 2
= 4π ∫ 4 4 − x 2 dx − 4π ∫ x 4 − x 2 dx Exercise 3C;
−2 −2
1, 3, 9, 12, 20
2 2
= 16π ∫ 4 − x 2 dx − 4π ∫ x 4 − x 2 dx
−2 −2
odd × even = odd function
semi-circle
= 16π  π ( 2)  − 0
1 2
2 
 
= 32π 2 units 3