Download - 11X1 T16 05 volumes (2011)
Volumes of Solids of Revolution
y
x
y = f(x)
Volumes of Solids of Revolution
y
x
y = f(x)
a b
Volumes of Solids of Revolution
y
x
y = f(x)
a b
Volumes of Solids of Revolution
y
x
y = f(x)
a b
Volume of a solid of revolution about;
Volumes of Solids of Revolution
y
x
y = f(x)
a b
b
adxyVxi 2:axis Volume of a solid of revolution about;
Volumes of Solids of Revolution
y
x
y = f(x)
a b
b
adxyVxi 2:axis Volume of a solid of revolution about;
d
cdyxVyii 2:axis
e.g. (i) cone
y
x
mxy
h
e.g. (i) cone
y
x
mxy
h
e.g. (i) cone
y
x
mxy
h
dxxm
dxyV
0
22
2
h
e.g. (i) cone
y
x
mxy
h
dxxm
dxyV
0
22
2
h
xm0
32
31
h
e.g. (i) cone
y
x
mxy
h
dxxm
dxyV
0
22
2
h
xm0
32
31
32
32
31
031
hm
hm
h
e.g. (i) cone
y
x
mxy
h
dxxm
dxyV
0
22
2
h
xm0
32
31
32
32
31
031
hm
hm
h
r
e.g. (i) cone
y
x
mxy
h
dxxm
dxyV
0
22
2
h
xm0
32
31
32
32
31
031
hm
hm
h
r
hrm
e.g. (i) cone
y
x
mxy
h
dxxm
dxyV
0
22
2
h
xm0
32
31
32
32
31
031
hm
hm
h
r
hrm
3
3
31
2
2
32
32
hrhhr
hhr
(ii) sphere
y
x
222 ryx
r
(ii) sphere
y
x
222 ryx
r
(ii) sphere
y
x
222 ryx
r
r
dxxr
dxyV
0
22
2
2
(ii) sphere
y
x
222 ryx
r
r
dxxr
dxyV
0
22
2
2
r
xxr0
32
312
(ii) sphere
y
x
222 ryx
r
r
dxxr
dxyV
0
22
2
2
r
xxr0
32
312
3
3
32
34
322
0312
r
r
rrr
(iii) .1 and 0between axis thearound
revolved is when generated solid theof volume theFind 2
yyyxy
(iii) .1 and 0between axis thearound
revolved is when generated solid theof volume theFind 2
yyyxy
y
x
2xy
(iii) .1 and 0between axis thearound
revolved is when generated solid theof volume theFind 2
yyyxy
y
x
2xy
1
(iii) .1 and 0between axis thearound
revolved is when generated solid theof volume theFind 2
yyyxy
y
x
2xy
1
(iii) .1 and 0between axis thearound
revolved is when generated solid theof volume theFind 2
yyyxy
y
x
2xy
1
1
0
2
ydy
dyxV
(iii) .1 and 0between axis thearound
revolved is when generated solid theof volume theFind 2
yyyxy
y
x
2xy
1
1
0
2
ydy
dyxV
102
2y
(iii) .1 and 0between axis thearound
revolved is when generated solid theof volume theFind 2
yyyxy
y
x
2xy
1
1
0
2
ydy
dyxV
102
2y
3
2
units 2
012
(iv)
y
x
xy 5
Find the volume of the solid when the shaded region is rotated about the x axis
5
(iv)
y
x
xy 5
1
0
2
1
0
22
2
2525
55
dxx
dxx
dxyV
Find the volume of the solid when the shaded region is rotated about the x axis
5
(iv)
y
x
xy 5
1
0
2
1
0
22
2
2525
55
dxx
dxx
dxyV
1
0
3
3125
xx
Find the volume of the solid when the shaded region is rotated about the x axis
5
(iv)
y
x
xy 5
1
0
2
1
0
22
2
2525
55
dxx
dxx
dxyV
1
0
3
3125
xx
3
3
unit 3
50
0131125
Find the volume of the solid when the shaded region is rotated about the x axis
5
(iv)
y
x
xy 5
1
0
2
1
0
22
2
2525
55
dxx
dxx
dxyV
1
0
3
3125
xx
3
3
unit 3
50
0131125
Find the volume of the solid when the shaded region is rotated about the x axis
5
OR
(iv)
y
x
xy 5
1
0
2
1
0
22
2
2525
55
dxx
dxx
dxyV
1
0
3
3125
xx
3
3
unit 3
50
0131125
Find the volume of the solid when the shaded region is rotated about the x axis
5
OR
3
2
22
units 3
50
1532
31
hrhrV
2005 HSC Question 6c)
The graphs of the curves and are shown in the diagram.
2y x 212 2y x
(i) Find the points of intersection of the two curves. (1)
2005 HSC Question 6c)
The graphs of the curves and are shown in the diagram.
2y x 212 2y x
(i) Find the points of intersection of the two curves. (1)2 212 2x x
2005 HSC Question 6c)
The graphs of the curves and are shown in the diagram.
2y x 212 2y x
(i) Find the points of intersection of the two curves. (1)2 212 2x x 2
2
3 124
2
xxx
2005 HSC Question 6c)
The graphs of the curves and are shown in the diagram.
2y x 212 2y x
(i) Find the points of intersection of the two curves. (1)2 212 2x x 2
2
3 124
2
xxx
meet at 2,4
(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 twoparts, or otherwise, find the volume of the solid formed.
(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 twoparts, or otherwise, find the volume of the solid formed.
(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 twoparts, or otherwise, find the volume of the solid formed.
2V x dy
(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 twoparts, or otherwise, find the volume of the solid formed.
2V x dy
(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 twoparts, or otherwise, find the volume of the solid formed.
2V x dy
2x y
(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 twoparts, or otherwise, find the volume of the solid formed.
2V x dy
2x y
2 162
x y
(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 twoparts, or otherwise, find the volume of the solid formed.
2V x dy
2x y
2 162
x y
4 12
0 4
162
V y dy ydy
(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 twoparts, or otherwise, find the volume of the solid formed.
2V x dy
2x y
2 162
x y
4 12
0 4
162
V y dy ydy
4 122 2
0 4
1 164 2
y y y
(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 twoparts, or otherwise, find the volume of the solid formed.
2V x dy
2x y
2 162
x y
4 12
0 4
162
V y dy ydy
4 122 2
0 4
1 164 2
y y y
2 2 216 4 4 0 12 44 2
(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 twoparts, or otherwise, find the volume of the solid formed.
2V x dy
2x y
2 162
x y
4 12
0 4
162
V y dy ydy
4 122 2
0 4
1 164 2
y y y
2 2 216 4 4 0 12 44 2
384 units
Exercise 11G; 3bdef, 4eg, 5cd, 6d, 8ad, 12, 16a, 19 to 22, 24*, 25*
