The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.

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
Volume of a solid of revolution about;
b
x

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

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

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

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

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

14. V    y 2 dx
e.g. (i) cone
h
y
y  mx
r
h x
r
m
h
   m 2 x 2 dx
0
h
2 1 3 
 m  x 
3  0
1 2 3
 m h  0 
3
1 2 3
 m h
3
2
1 r
    h3
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
x2  y2  r 2
V    y 2 dx
r
 2  r 2  x 2 dx
0
rx

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

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

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  x2
y
V 

1

x 2 dy
1
 ydy
0

x

y 
2
2 1
0

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  x2
y
V 

1

x 2 dy
1
 ydy
0

x



y 
2

2

2
2 1
0
1
2
 0
units
3

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
2
   52  5 x  dx


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
2
   52  5 x  dx


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
2
   52  5 x  dx


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
2
   52  5 x  dx


0
1
x
OR
   25  25 x 2 dx
0
1
 x  1 x3 
 25 
 3 0

1 1 13 0
 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
2
   52  5 x  dx


0
1
x
1
OR V  r 2 h  r 2 h
3
2
2
  5 1
3
50

units 3
3
   25  25 x 2 dx
0
1
 x  1 x3 
 25 
 3 0

1 1 13 0
 25  
 
 3

50

unit 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.
x 2  12  2 x 2
(1)

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.
x 2  12  2 x 2
3 x 2  12
x2  4
x  2
(1)

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.
x 2  12  2 x 2
3 x 2  12
x2  4
x  2
 meet at  2, 4 
(1)

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
2
V    x 2 dy

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
2
V    x 2 dy
4
12
  1  
V   6
y  dy
 ydy
2 
0
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
2
V    x 2 dy
4
12
  1  
V   6
y  dy
 ydy
2 
0
4
4
12
1 2
1 2
  6 y  y     y 

 2 4
4 0



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
2
V    x 2 dy
4
12


  1  
V   6
y  dy
 ydy
2 
0
4
4
12
1 2
1 2
  6 y  y     y 

 2 4
4 0


1 2

2
2
  6  4    4   0  12    4 
4
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
2
V    x 2 dy
4
12


  1  
V   6
y  dy
 ydy
2 
0
4
4
12
1 2
1 2
  6 y  y     y 

 2 4
4 0


1 2

2
2
  6  4    4   0  12    4 
4
2
 84 units3



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