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rotation || slice
Volumes By Cylindrical Shells
rotation || slice
Volumes By Cylindrical Shells
y
x
xfy
rotation || slice
Volumes By Cylindrical Shells
y
x
xfy
a b
rotation || slice
Volumes By Cylindrical Shells
y
x
xfy
a x b
rotation || slice
Volumes By Cylindrical Shells
y
x
xfy
a x b
rotation || slice
Volumes By Cylindrical Shells
y
x
xfy
a x b
rotation || slice
Volumes By Cylindrical Shells
y
x
xfy
a xx
b
rotation || slice
Volumes By Cylindrical Shells
y
x
xfy
a x
y
xb
rotation || slice
Volumes By Cylindrical Shells
y
x
xfy
a x
y
x
xf
b
rotation || slice
Volumes By Cylindrical Shells
y
x
xfy
a x
y
x
xf
x2b
rotation || slice
Volumes By Cylindrical Shells
y
x
xfy
a x
y
x
xf
xfxxA 2
x2b
rotation || slice
Volumes By Cylindrical Shells
y
x
xfy
a x
y
x
xf
xfxxA 2
xxfxV 2
x2b
rotation || slice
Volumes By Cylindrical Shells
y
x
xfy
a x
y
x
xf
xfxxA 2
xxfxV 2
b
axx
xxfxV 2lim0
x2b
rotation || slice
Volumes By Cylindrical Shells
y
x
xfy
a x
y
x
xf
xfxxA 2
xxfxV 2
b
axx
xxfxV 2lim0
b
a
dxxxf2
x2b
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 2
y
y4
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2
y
y4
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2
y
y4
y
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2
y
y4
y
x2
2
1
4 yx
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2
y
y4
y
x2
2
1
4 yx
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2
y
y4
y
x2
2
1
4 yx
2
1
42 y
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2
y
y4
y
x2
2
1
4 yx
2
1
42 y
y2
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2
y
y4
y
x2
2
1
422 yyyA
2
1
4 yx
2
1
42 y
y2
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2
y
y4
y
x2
2
1
422 yyyA
yyyV 2
1
44
2
1
4 yx
2
1
42 y
y2
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2
y
y4
y
x2
2
1
422 yyyA
yyyV 2
1
44
4
0
2
1
044lim
xy
yyyV
2
1
4 yx
2
1
42 y
y2
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2
y
y4
y
x2
2
1
422 yyyA
yyyV 2
1
44
4
0
2
1
044lim
xy
yyyV
4
0
2
1
44 dyyy 2
1
4 yx
2
1
42 y
y2
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2
y
y4
y
x2
2
1
422 yyyA
yyyV 2
1
44
4
0
2
1
044lim
xy
yyyV
4
0
2
1
44 dyyy 2
1
4 yx
2
1
42 y
y2
4
0
2
1
44 dyyy
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2
y
y4
y
x2
2
1
422 yyyA
yyyV 2
1
44
4
0
2
1
044lim
xy
yyyV
4
0
2
1
44 dyyy 2
1
4 yx
2
1
42 y
y2
4
0
2
1
44 dyyy
4
0
2
3
2
1
44 dyyy
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2
y
y4
y
x2
2
1
422 yyyA
yyyV 2
1
44
4
0
2
1
044lim
xy
yyyV
4
0
2
1
44 dyyy
4
0
2
5
2
3
52
38
4
yy
2
1
4 yx
2
1
42 y
y2
4
0
2
1
44 dyyy
4
0
2
3
2
1
44 dyyy
axis about the rotated is axis the
and 4between area when thegenerated volume theFind e.g. 2
xx
xyi
x
24 xy
-2 2
y
y4
y
x2
2
1
422 yyyA
yyyV 2
1
44
4
0
2
1
044lim
xy
yyyV
4
0
2
1
44 dyyy
4
0
2
5
2
3
52
38
4
yy
3units 15
512
03252
838
4
2
1
4 yx
2
1
42 y
y2
4
0
2
1
44 dyyy
4
0
2
3
2
1
44 dyyy
4 line about the
rotated is 4 when generated volume theFind 22
x
yxii
4 line about the
rotated is 4 when generated volume theFind 22
x
yxii
x
422 yx
y
2
-2
2-2
4 line about the
rotated is 4 when generated volume theFind 22
x
yxii
x
422 yx
4
y
2
-2
2-2
4 line about the
rotated is 4 when generated volume theFind 22
x
yxii
x
422 yx
4
x
y
2
-2
2-2
4 line about the
rotated is 4 when generated volume theFind 22
x
yxii
x
422 yx
4
x
y
2
-2
2-2
4 line about the
rotated is 4 when generated volume theFind 22
x
yxii
x
422 yx
4
x
y
2
-2
2-2
4 line about the
rotated is 4 when generated volume theFind 22
x
yxii
x
422 yx
4
x
y
2
-2
2-2
torus
4 line about the
rotated is 4 when generated volume theFind 22
x
yxii
x
422 yx
4
x
y
2
-2
2-2
torus
4 line about the
rotated is 4 when generated volume theFind 22
x
yxii
x
422 yx
4
x
y
2
-2
2-2
torus
x
4 line about the
rotated is 4 when generated volume theFind 22
x
yxii
x
422 yx
4
x
y
2
-2
2-2
torus
x
y2
4 line about the
rotated is 4 when generated volume theFind 22
x
yxii
x
422 yx
4
x
y
2
-2
2-2
torus
x
y2242 x
4 line about the
rotated is 4 when generated volume theFind 22
x
yxii
x
422 yx
4
x
y
2
-2
2-2
torus
x
y2
x42
242 x
4 line about the
rotated is 4 when generated volume theFind 22
x
yxii
x
422 yx
4
x
y
2
-2
2-2
torus
x
y2
x42
242 x
xxxA 4242 2
4 line about the
rotated is 4 when generated volume theFind 22
x
yxii
x
422 yx
4
x
y
2
-2
2-2
torus
x
y2
x42
242 x
xxxA 4242 2
2444 xx
4 line about the
rotated is 4 when generated volume theFind 22
x
yxii
x
422 yx
4
x
y
2
-2
2-2
torus
x
y2
x42
242 x
xxxA 4242 2
xxxV 2444
2444 xx
2
2
2
0444lim
xx
xxxV
2
2
2
0444lim
xx
xxxV
2
2
2444 dxxx
2
2
2
0444lim
xx
xxxV
2
2
2444 dxxx
2
2
22
2
2 44444 dxxxdxx
2
2
2
0444lim
xx
xxxV
2
2
2444 dxxx
2
2
22
2
2 44444 dxxxdxx
2
2
22
2
2 44416 dxxxdxx
2
2
2
0444lim
xx
xxxV
2
2
2444 dxxx
2
2
22
2
2 44444 dxxxdxx
2
2
22
2
2 44416 dxxxdxx
semi-circle
2
2
2
0444lim
xx
xxxV
2
2
2444 dxxx
2221
16
2
2
22
2
2 44444 dxxxdxx
2
2
22
2
2 44416 dxxxdxx
semi-circle
2
2
2
0444lim
xx
xxxV
2
2
2444 dxxx
2221
16
2
2
22
2
2 44444 dxxxdxx
2
2
22
2
2 44416 dxxxdxx
semi-circle function odd even odd
2
2
2
0444lim
xx
xxxV
2
2
2444 dxxx
2221
16
2
2
22
2
2 44444 dxxxdxx
2
2
22
2
2 44416 dxxxdxx
semi-circle function odd even odd
0
2
2
2
0444lim
xx
xxxV
2
2
2444 dxxx
2221
16
2
2
22
2
2 44444 dxxxdxx
2
2
22
2
2 44416 dxxxdxx
semi-circle function odd even odd
0
32units32
2
2
2
0444lim
xx
xxxV
2
2
2444 dxxx
2221
16
Exercise 3B; 3, 4, 6, 8
Exercise 3C;1, 3, 9, 12, 20
2
2
22
2
2 44444 dxxxdxx
2
2
22
2
2 44416 dxxxdxx
semi-circle function odd even odd
0
32units32