Volumes of Solids rotation slice Volumes By Discs & Washers
Jun 04, 2015
Volumes of Solids rotationslice
Volumes By Discs & Washers
Volumes of Solids rotationslice
About the x axis
Volumes By Discs & Washers
Volumes of Solids rotationslice
About the x axis
Volumes By Discs & Washersy
x
xfy
Volumes of Solids rotationslice
About the x axis
Volumes By Discs & Washersy
x
xfy
a b
Volumes of Solids rotationslice
About the x axis
Volumes By Discs & Washersy
x
xfy
a bx
Volumes of Solids rotationslice
About the x axis
Volumes By Discs & Washersy
x
xfy
a bx
y
x
Volumes of Solids rotationslice
About the x axis
Volumes By Discs & Washersy
x
xfy
a bx
y
x
xf
Volumes of Solids rotationslice
About the x axis
Volumes By Discs & Washersy
x
xfy
a bx
y
x
xf
2xfxA
Volumes of Solids rotationslice
About the x axis
Volumes By Discs & Washersy
x
xfy
a bx
y
x
xf
2xfxA
xxfV 2
Volumes of Solids rotationslice
About the x axis
Volumes By Discs & Washersy
x
xfy
a bx
y
x
xf
2xfxA
xxfV 2
b
axxxxfV 2
0lim
Volumes of Solids rotationslice
About the x axis
Volumes By Discs & Washersy
x
xfy
a bx
y
x
xf
2xfxA
xxfV 2
b
axxxxfV 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 xy
About the y axisy
x
xfy
c
dy xy yg
About the y axisy
x
xfy
c
dy xy yg
2ygyA
About the y axisy
x
xfy
c
dy xy yg
2ygyA
yygV 2
About the y axisy
x
xfy
c
dy xy yg
2ygyA
yygV 2
d
cyyyygV 2
0lim
About the y axisy
x
xfy
c
dy xy yg
2ygyA
yygV 2
d
cyyyygV 2
0lim
d
c
dyyg 2
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2
y4
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2x
y4
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2x
y4
x
24 xy
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2x
y4
x
24 xy
224 xxA
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2x
y4
x
24 xy
224 xxA xxxV 42816
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2x
y4
x
24 xy
224 xxA xxxV 42816
2
2
42
0816lim
xxxxxV
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2x
y4
x
24 xy
224 xxA xxxV 42816
2
2
42
0816lim
xxxxxV
2
0
428162 dxxx
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2x
y4
x
24 xy
224 xxA xxxV 42816
2
2
42
0816lim
xxxxxV
2
0
428162 dxxx
2
0
53
51
38162
xxx
axisabout therotatedis axis the
and 4between area when thegenerated volume theFind e.g. 2
xxxyi
x
24 xy
-2 2x
y4
x
24 xy
224 xxA xxxV 42816
2
2
42
0816lim
xxxxxV
2
0
428162 dxxx
2
0
53
51
38162
xxx
3units 15
512
05
323
64322
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yyxyii
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yyxyii
x24 xy
y4
3y
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yyxyii
x24 xy
(-1,3) (1,3)
y4
3y
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yyxyii
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
yyxyii
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
yyxyii
x24 xy
(-1,3) (1,3)x
y4
x
2
2
1343
xxy
3y
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yyxyii
x24 xy
(-1,3) (1,3)x
y4
x
2
2
1343
xxy
221 xxA
3y
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yyxyii
x24 xy
(-1,3) (1,3)x
y4
x
2
2
1343
xxy
221 xxA
xxxV 4221
3y
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yyxyii
x24 xy
(-1,3) (1,3)x
y4
x
2
2
1343
xxy
221 xxA
xxxV 4221
1
1
42
021lim
xxxxxV
3y
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yyxyii
x24 xy
(-1,3) (1,3)x
y4
x
2
2
1343
xxy
221 xxA
xxxV 4221
1
1
42
021lim
xxxxxV
1
0
42212 dxxx3y
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yyxyii
x24 xy
(-1,3) (1,3)x
y4
x
2
2
1343
xxy
221 xxA
xxxV 4221
1
1
42
021lim
xxxxxV
1
0
42212 dxxx
1
0
53
51
322
xxx
3y
3 line about the rotated is 3
and 4by enclosed area when thegenerated volume theFind 2
yyxyii
x24 xy
(-1,3) (1,3)x
y4
x
2
2
1343
xxy
221 xxA
xxxV 4221
1
1
42
021lim
xxxxxV
1
0
42212 dxxx
1
0
53
51
322
xxx
3units 15
16
051
3212
3y
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
yyxxyiii
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
yyxxyiii
y
x
2xy 2yx
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
yyxxyiii
y
x
2xy
(1,1)
2yx
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
yyxxyiii
y
x
2xy
(1,1)y
2yx
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
yyxxyiii
y
x
2xy
(1,1)y
2yx
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
yyxxyiii
y
x
2xy
(1,1)y
2yx
21
yx
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
yyxxyiii
y
x
2xy
(1,1)y
2yx
21
yx 2yx
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
yyxxyiii
y
x
2xy
(1,1)y
2yx
21
yx 2yx
222
21
yyyA
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
yyxxyiii
y
x
2xy
(1,1)y
2yx
21
yx 2yx
222
21
yyyA
yyyV 4
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
yyxxyiii
y
x
2xy
(1,1)y
2yx
21
yx 2yx
222
21
yyyA
yyyV 4
1
0
4
0lim
yyyyyV
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
yyxxyiii
y
x
2xy
(1,1)y
2yx
21
yx 2yx
222
21
yyyA
yyyV 4
1
0
4
0lim
yyyyyV
1
0
4 dyyy
generated. volume theFind axis. line about the
rotated is and curves ebetween th area The 22
yyxxyiii
y
x
2xy
(1,1)y
2yx
21
yx 2yx
222
21
yyyA
yyyV 4
1
0
4
0lim
yyyyyV
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
yyxxyiii
y
x
2xy
(1,1)y
2yx
21
yx 2yx
222
21
yyyA
yyyV 4
1
0
4
0lim
yyyyyV
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 r3
14
x1 4
1
-1
S02 yx r3
14
R
x1 4
1
-1
S02 yx r3
14
R24
4yx
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
4yx
a) Show that the area of the annulus at height y is equal to 78 24 yy
y
r
R24
4yx
314
a) Show that the area of the annulus at height y is equal to 78 24 yy
y
r
R24
4yx
314
222 34 yyA
a) Show that the area of the annulus at height y is equal to 78 24 yy
y
r
R24
4yx
314
222 34 yyA
78
981624
42
yyyy
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
yyyyyV
b) Hence find the volume of the solid.
yyyV 78 24
1
1
24
078lim
yyyyyV
1
0
24 782 dyyy
b) Hence find the volume of the solid.
yyyV 78 24
1
1
24
078lim
yyyyyV
1
0
24 782 dyyy
1
0
35 738
512
yyy
b) Hence find the volume of the solid.
yyyV 78 24
1
1
24
078lim
yyyyyV
1
0
24 782 dyyy
1
0
35 738
512
yyy
3units 15
296
0738
512
b) Hence find the volume of the solid.
yyyV 78 24
1
1
24
078lim
yyyyyV
1
0
24 782 dyyy
1
0
35 738
512
yyy
3units 15
296
0738
512
Exercise 3A; 2, 5, 7, 9, 13, 14, 15
Exercise 3B;1, 2, 5, 7, 9, 10, 11, 12