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Volumes of Solids rotation slice
Volumes By Discs & Washers
Volumes of Solids rotation slice
About the x axis
Volumes By Discs & Washers
Volumes of Solids rotation slice
About the x axis
Volumes By Discs & Washers
y
x
xfy
Volumes of Solids rotation slice
About the x axis
Volumes By Discs & Washers
y
x
xfy
a b
Volumes of Solids rotation slice
About the x axis
Volumes By Discs & Washers
y
x
xfy
a bx
Volumes of Solids rotation slice
About the x axis
Volumes By Discs & Washers
y
x
xfy
a bx
y
x
Volumes of Solids rotation slice
About the x axis
Volumes By Discs & Washers
y
x
xfy
a bx
y
x
xf
Volumes of Solids rotation slice
About the x axis
Volumes By Discs & Washers
y
x
xfy
a bx
y
x
xf
2xfxA
Volumes of Solids rotation slice
About the x axis
Volumes By Discs & Washers
y
x
xfy
a bx
y
x
xf
2xfxA
xxfV 2
Volumes of Solids rotation slice
About the x axis
Volumes By Discs & Washers
y
x
xfy
a bx
y
x
xf
2xfxA
xxfV 2
b
axx
xxfV 2
0lim
Volumes of Solids rotation slice
About the x axis
Volumes By Discs & Washers
y
x
xfy
a bx
y
x
xf
2xfxA
xxfV 2
b
axx
xxfV 2
0lim
b
a
dxxf 2
About the y axisy
x
xfy
About the y axisy
x
xfy
c
d
About the y axisy
x
xfy
c
dy
About the y axisy
x
xfy
c
dy x
y
About the y axisy
x
xfy
c
dy x
y yg
About the y axisy
x
xfy
c
dy x
y yg
2ygyA
About the y axisy
x
xfy
c
dy x
y yg
2ygyA
yygV 2
About the y axisy
x
xfy
c
dy x
y yg
2ygyA
yygV 2
d
cyy
yygV 2
0lim
About the y axisy
x
xfy
c
dy x
y yg
2ygyA
yygV 2
d
cyy
yygV 2
0lim
d
c
dyyg 2
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2
y4
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2x
y4
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2x
y4
x
24 xy
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2x
y4
x
24 xy
224 xxA
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2x
y4
x
24 xy
224 xxA xxxV 42816
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2x
y4
x
24 xy
224 xxA xxxV 42816
2
2
42
0816lim
xx
xxxV
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2x
y4
x
24 xy
224 xxA xxxV 42816
2
2
42
0816lim
xx
xxxV
2
0
428162 dxxx
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2x
y4
x
24 xy
224 xxA xxxV 42816
2
2
42
0816lim
xx
xxxV
2
0
428162 dxxx2
0
53
51
38
162
xxx
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2x
y4
x
24 xy
224 xxA xxxV 42816
2
2
42
0816lim
xx
xxxV
2
0
428162 dxxx2
0
53
51
38
162
xxx
3units 15
512
05
323
64322
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yy
xyii
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yy
xyii
x24 xy
y4
3y
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yy
xyii
x24 xy
(-1,3) (1,3)
y4
3y
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yy
xyii
x24 xy
(-1,3) (1,3)x
y4
3y
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yy
xyii
x24 xy
(-1,3) (1,3)x
y4
x
3y
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yy
xyii
x24 xy
(-1,3) (1,3)x
y4
x
2
2
1
343
x
xy
3y
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yy
xyii
x24 xy
(-1,3) (1,3)x
y4
x
2
2
1
343
x
xy
221 xxA
3y
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yy
xyii
x24 xy
(-1,3) (1,3)x
y4
x
2
2
1
343
x
xy
221 xxA
xxxV 4221
3y
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yy
xyii
x24 xy
(-1,3) (1,3)x
y4
x
2
2
1
343
x
xy
221 xxA
xxxV 4221
1
1
42
021lim
xx
xxxV
3y
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yy
xyii
x24 xy
(-1,3) (1,3)x
y4
x
2
2
1
343
x
xy
221 xxA
xxxV 4221
1
1
42
021lim
xx
xxxV
1
0
42212 dxxx3y
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yy
xyii
x24 xy
(-1,3) (1,3)x
y4
x
2
2
1
343
x
xy
221 xxA
xxxV 4221
1
1
42
021lim
xx
xxxV
1
0
42212 dxxx1
0
53
51
32
2
xxx
3y
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yy
xyii
x24 xy
(-1,3) (1,3)x
y4
x
2
2
1
343
x
xy
221 xxA
xxxV 4221
1
1
42
021lim
xx
xxxV
1
0
42212 dxxx1
0
53
51
32
2
xxx
3units 15
16
051
32
12
3y
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
y
yxxyiii
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
y
yxxyiii
y
x
2xy
2yx
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
y
yxxyiii
y
x
2xy
(1,1)
2yx
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
y
yxxyiii
y
x
2xy
(1,1)y
2yx
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
y
yxxyiii
y
x
2xy
(1,1)y
2yx
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
y
yxxyiii
y
x
2xy
(1,1)y
2yx
2
1
yx
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
y
yxxyiii
y
x
2xy
(1,1)y
2yx
2
1
yx 2yx
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
y
yxxyiii
y
x
2xy
(1,1)y
2yx
2
1
yx 2yx
22
2
2
1
yyyA
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
y
yxxyiii
y
x
2xy
(1,1)y
2yx
2
1
yx 2yx
22
2
2
1
yyyA
yyyV 4
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
y
yxxyiii
y
x
2xy
(1,1)y
2yx
2
1
yx 2yx
22
2
2
1
yyyA
yyyV 4
1
0
4
0lim
yy
yyyV
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
y
yxxyiii
y
x
2xy
(1,1)y
2yx
2
1
yx 2yx
22
2
2
1
yyyA
yyyV 4
1
0
4
0lim
yy
yyyV
1
0
4 dyyy
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
y
yxxyiii
y
x
2xy
(1,1)y
2yx
2
1
yx 2yx
22
2
2
1
yyyA
yyyV 4
1
0
4
0lim
yy
yyyV
1
0
4 dyyy
1
0
52
51
21
yy
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
y
yxxyiii
y
x
2xy
(1,1)y
2yx
2
1
yx 2yx
22
2
2
1
yyyA
yyyV 4
1
0
4
0lim
yy
yyyV
1
0
4 dyyy
1
0
52
51
21
yy
3units 103
051
21
(iv) (1996) y
x1 4
1
-1
S02 yx
The shaded region is bounded by the lines x = 1, y = 1 and y = -1 and the curve . The region is rotated through about the line x = 4 to form a solid.
When the region is rotated, the line segment S at height y sweeps out an annulus.
02 yx 360
a) Show that the area of the annulus at height y is equal to 78 24 yy
a) Show that the area of the annulus at height y is equal to 78 24 yy
y
a) Show that the area of the annulus at height y is equal to 78 24 yy
y r
a) Show that the area of the annulus at height y is equal to 78 24 yy
y
R
x1 4
1
-1
S02 yx
x1 4
1
-1
S02 yx r
x1 4
1
-1
S02 yx r
3
14
x1 4
1
-1
S02 yx r
3
14
R
x1 4
1
-1
S02 yx r
3
14
R24
4
y
x
a) Show that the area of the annulus at height y is equal to 78 24 yy
y
a) Show that the area of the annulus at height y is equal to 78 24 yy
y
R24
4
y
x
a) Show that the area of the annulus at height y is equal to 78 24 yy
y
r
R24
4
y
x
3
14
a) Show that the area of the annulus at height y is equal to 78 24 yy
y
r
R24
4
y
x
3
14
222 34 yyA
a) Show that the area of the annulus at height y is equal to 78 24 yy
y
r
R24
4
y
x
3
14
222 34 yyA
78
981624
42
yy
yy
b) Hence find the volume of the solid.
b) Hence find the volume of the solid.
yyyV 78 24
b) Hence find the volume of the solid.
yyyV 78 24
1
1
24
078lim
yy
yyyV
b) Hence find the volume of the solid.
yyyV 78 24
1
1
24
078lim
yy
yyyV
1
0
24 782 dyyy
b) Hence find the volume of the solid.
yyyV 78 24
1
1
24
078lim
yy
yyyV
1
0
24 782 dyyy
1
0
35 738
51
2
yyy
b) Hence find the volume of the solid.
yyyV 78 24
1
1
24
078lim
yy
yyyV
1
0
24 782 dyyy
1
0
35 738
51
2
yyy
3units 15
296
0738
51
2
b) Hence find the volume of the solid.
yyyV 78 24
1
1
24
078lim
yy
yyyV
1
0
24 782 dyyy
1
0
35 738
51
2
yyy
3units 15
296
0738
51
2
Exercise 3A; 2, 5, 7, 9, 13, 14, 15
Exercise 3B;1, 2, 5, 7, 9, 10, 11, 12