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
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
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 13 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 13 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 13 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