1) The document discusses calculating volumes of solids formed by rotating areas bounded by graphs around axes. 2) Formulas are provided to calculate volumes when rotating around the x-axis (V = π ∫y2 dx) or y-axis (V = π ∫x2 dy). 3) Examples are worked out for rotating a cone around the x-axis and a sphere around both axes to derive their volume formulas.

1. Volumes of Solids
of Revolution
y
y = f(x)
x

2. Volumes of Solids
of Revolution
y
y = f(x)
a b x

3. Volumes of Solids
of Revolution
y
y = f(x)
a b x

4. Volumes of Solids
of Revolution
y
y = f(x)
a b x
Volume of a solid of revolution about;

5. Volumes of Solids
of Revolution
y
y = f(x)
a b x
Volume of a solid of revolution about; i x axis : V   y 2 dx
b
 a

6. Volumes of Solids
of Revolution
y
y = f(x)
a b x
Volume of a solid of revolution about; i x axis : V   y 2 dx
b
 a
ii  y axis : V   c x 2 dy
d

7. e.g. (i) cone
y
y  mx
h x

8. e.g. (i) cone
y
y  mx
h x

9. e.g. (i) cone V    y 2 dx
h
y    m 2 x 2 dx
y  mx 0
h x

10. e.g. (i) cone V    y 2 dx
h
y    m 2 x 2 dx
y  mx 0
h
2 1 3 
 m  x 
3  0
h x

11. e.g. (i) cone V    y 2 dx
h
y    m 2 x 2 dx
y  mx 0
h
2 1 3 
 m  x 
3  0
 m 2 h 3  0 
1
h x 3
1
 m 2 h 3
3

12. e.g. (i) cone V    y 2 dx
h
y    m 2 x 2 dx
y  mx 0
h
2 1 3 
 m  x 
3  0
r
 m 2 h 3  0 
1
h x 3
1
 m 2 h 3
3

13. e.g. (i) cone V    y 2 dx
h
y    m 2 x 2 dx
y  mx 0
h
2 1 3 
 m  x 
3  0
r
 m 2 h 3  0 
1
h x 3
1
 m 2 h 3
3
r
m
h

14. e.g. (i) cone V    y 2 dx
h
y    m 2 x 2 dx
y  mx 0
h
2 1 3 
 m  x 
3  0
r
 m 2 h 3  0 
1
h x 3
1
 m 2 h 3
3 2
r
    h3
m 1 r
 
h 3 h
r 2 h 3

3h 2
r 2 h

3

15. (ii) sphere
y
x2  y2  r 2
rx

16. (ii) sphere
y
x2  y2  r 2
rx

17. (ii) sphere
y V    y 2 dx
x2  y2  r 2 r
 2  r 2  x 2 dx
0
rx

18. (ii) sphere
y V    y 2 dx
x2  y2  r 2 r
 2  r 2  x 2 dx
0
r
r 2 x  1 x 3 
 2 
rx
 3 0 

19. (ii) sphere
y V    y 2 dx
x2  y2  r 2 r
 2  r 2  x 2 dx
0
r
r 2 x  1 x 3 
 2 
rx
 3 0 
 1  
 2  r 2 r   r 3   0

 3  
2 3
 2  r 
3 
4 3
 r
3

20. (iii) Find the volume of the solid generated when y  x 2 is revolved
around the y axis between y  0 and y  1.

21. (iii) Find the volume of the solid generated when y  x 2 is revolved
around the y axis between y  0 and y  1.
y y  x2
x

22. (iii) Find the volume of the solid generated when y  x 2 is revolved
around the y axis between y  0 and y  1.
y y  x2
1
x

23. (iii) Find the volume of the solid generated when y  x 2 is revolved
around the y axis between y  0 and y  1.
y y  x2
1
x

24. (iii) Find the volume of the solid generated when y  x 2 is revolved
around the y axis between y  0 and y  1.
y y  x2 V   x 2 dy
1
1   ydy
0
x

25. (iii) Find the volume of the solid generated when y  x 2 is revolved
around the y axis between y  0 and y  1.
y y  x2 V   x 2 dy
1
1   ydy
0


2
y  2 1
0
x

26. (iii) Find the volume of the solid generated when y  x 2 is revolved
around the y axis between y  0 and y  1.
y y  x2 V   x 2 dy
1
1   ydy
0


2
y  2 1
0
x

 1 2
 0
2

 units 3
2

27. (iv) Find the volume of the solid when the shaded region is rotated
about the x axis
y
y  5x
5
x

28. (iv) Find the volume of the solid when the shaded region is rotated
about the x axis
y V    y 2 dx
y  5x
 
1
5
   52  5 x  dx
2
0
1
x    25  25 x 2 dx
0

29. (iv) Find the volume of the solid when the shaded region is rotated
about the x axis
y V    y 2 dx
y  5x
 
1
5
   52  5 x  dx
2
0
1
x    25  25 x 2 dx
0
1
 x  1 x3 
 25 
 3 0 

30. (iv) Find the volume of the solid when the shaded region is rotated
about the x axis
y V    y 2 dx
y  5x
 
1
5
   52  5 x  dx
2
0
1
x    25  25 x 2 dx
0
1
 x  1 x3 
 25 
 3 0 
1 1 13 0
 25    
 3 
50
 unit 3
3

31. (iv) Find the volume of the solid when the shaded region is rotated
about the x axis
y V    y 2 dx
y  5x
 
1
5
   52  5 x  dx
2
0
1
x    25  25 x 2 dx
0
1
 x  1 x3 
 25 
 3 0 
1 1 13 0
OR  25    
 3 
50
 unit 3
3

32. (iv) Find the volume of the solid when the shaded region is rotated
about the x axis
y V    y 2 dx
y  5x
 
1
5
   52  5 x  dx
2
0
1
x    25  25 x 2 dx
0
1
 x  1 x3 
 25 
 3 0 
1 1 1 13 0
OR V  r 2 h  r 2 h  25    
3  3 
2 50
  5 1 
2
unit 3
3 3
50
 units 3
3

33. 2005 HSC Question 6c)
The graphs of the curves y  x 2 and y  12  2 x 2 are shown in the
diagram.
(i) Find the points of intersection of the two curves. (1)

34. 2005 HSC Question 6c)
The graphs of the curves y  x 2 and y  12  2 x 2 are shown in the
diagram.
(i) Find the points of intersection of the two curves. (1)
x 2  12  2 x 2

35. 2005 HSC Question 6c)
The graphs of the curves y  x 2 and y  12  2 x 2 are shown in the
diagram.
(i) Find the points of intersection of the two curves. (1)
x 2  12  2 x 2
3 x 2  12
x2  4
x  2

36. 2005 HSC Question 6c)
The graphs of the curves y  x 2 and y  12  2 x 2 are shown in the
diagram.
(i) Find the points of intersection of the two curves. (1)
x 2  12  2 x 2
3 x 2  12
x2  4
x  2
 meet at  2, 4 

37. (ii) The shaded region between the two curves and the y axis is (3)
rotated about the y axis. By splitting the shaded region into two
parts, or otherwise, find the volume of the solid formed.

38. (ii) The shaded region between the two curves and the y axis is (3)
rotated about the y axis. By splitting the shaded region into two
parts, or otherwise, find the volume of the solid formed.

39. (ii) The shaded region between the two curves and the y axis is (3)
rotated about the y axis. By splitting the shaded region into two
parts, or otherwise, find the volume of the solid formed.
V    x 2 dy

40. (ii) The shaded region between the two curves and the y axis is (3)
rotated about the y axis. By splitting the shaded region into two
parts, or otherwise, find the volume of the solid formed.
V    x 2 dy

41. (ii) The shaded region between the two curves and the y axis is (3)
rotated about the y axis. By splitting the shaded region into two
parts, or otherwise, find the volume of the solid formed.
x2  y
V    x 2 dy

42. (ii) The shaded region between the two curves and the y axis is (3)
rotated about the y axis. By splitting the shaded region into two
parts, or otherwise, find the volume of the solid formed.
x2  y
1
x  6 y
2
V    x 2 dy 2

43. (ii) The shaded region between the two curves and the y axis is (3)
rotated about the y axis. By splitting the shaded region into two
parts, or otherwise, find the volume of the solid formed.
x2  y
1
x  6 y
2
V    x 2 dy 2
4 12
  1  
V   6 y  dy  ydy
0
2  4

44. (ii) The shaded region between the two curves and the y axis is (3)
rotated about the y axis. By splitting the shaded region into two
parts, or otherwise, find the volume of the solid formed.
x2  y
1
x  6 y
2
V    x 2 dy 2
4 12
  1  
V   6 y  dy  ydy
0
2  4
4 12
  6 y  y     y 
1 2 1 2

 4 0  2 4


45. (ii) The shaded region between the two curves and the y axis is (3)
rotated about the y axis. By splitting the shaded region into two
parts, or otherwise, find the volume of the solid formed.
x2  y
1
x  6 y
2
V    x 2 dy 2
4 12
  1  
V   6 y  dy  ydy
0
2  4
4 12
  6 y  y     y 
1 2 1 2

 4 0  2 4

 1 2
 

  6  4    4   0  12    4 
4 2
2 2


46. (ii) The shaded region between the two curves and the y axis is (3)
rotated about the y axis. By splitting the shaded region into two
parts, or otherwise, find the volume of the solid formed.
x2  y
1
x  6 y
2
V    x 2 dy 2
4 12
  1  
V   6 y  dy  ydy
0
2  4
4 12
  6 y  y     y 
1 2 1 2

 4 0  2 4

 1 2
 

  6  4    4   0  12    4 
4 2
2 2

 84 units3

47. Exercise 11G; 3bdef, 4eg, 5cd, 6d, 8ad,
12, 16a, 19 to 22, 24*, 25*