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Volumes By Cylindrical Shells

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Volumes By Cylindrical Shellsy

x

xfy

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Volumes By Cylindrical Shellsy

x

xfy

a b

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Volumes By Cylindrical Shellsy

x

xfy

a x b

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Volumes By Cylindrical Shellsy

x

xfy

a x b

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Volumes By Cylindrical Shellsy

x

xfy

a x b

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Volumes By Cylindrical Shellsy

x

xfy

a xx

b

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Volumes By Cylindrical Shellsy

x

xfy

a x

y

xb

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Volumes By Cylindrical Shellsy

x

xfy

a x

y

x

xf

b

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Volumes By Cylindrical Shellsy

x

xfy

a x

y

x

xf

x2b

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Volumes By Cylindrical Shellsy

x

xfy

a x

y

x

xf

xfxxA 2

x2b

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Volumes By Cylindrical Shellsy

x

xfy

a x

y

x

xf

xfxxA 2 xxfxV 2

x2b

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Volumes By Cylindrical Shellsy

x

xfy

a x

y

x

xf

xfxxA 2 xxfxV 2

b

axxxxfxV 2lim

0

x2b

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Volumes By Cylindrical Shellsy

x

xfy

a x

y

x

xf

xfxxA 2 xxfxV 2

b

axxxxfxV 2lim

0

b

a

dxxxf2

x2b

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

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axisabout therotatedis axis the

and 4between area when thegenerated volume theFind e.g. 2

xxxyi

x

24 xy

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y

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axisabout therotatedis axis the

and 4between area when thegenerated volume theFind e.g. 2

xxxyi

x

24 xy

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y

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axisabout therotatedis axis the

and 4between area when thegenerated volume theFind e.g. 2

xxxyi

x

24 xy

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y

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axisabout therotatedis axis the

and 4between area when thegenerated volume theFind e.g. 2

xxxyi

x

24 xy

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y

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y

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axisabout therotatedis axis the

and 4between area when thegenerated volume theFind e.g. 2

xxxyi

x

24 xy

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axisabout therotatedis axis the

and 4between area when thegenerated volume theFind e.g. 2

xxxyi

x

24 xy

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axisabout therotatedis axis the

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xxxyi

x

24 xy

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4 yx

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axisabout therotatedis axis the

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axisabout therotatedis axis the

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axisabout therotatedis axis the

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0

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axisabout therotatedis axis the

and 4between area when thegenerated volume theFind e.g. 2

xxxyi

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axisabout therotatedis axis the

and 4between area when thegenerated volume theFind e.g. 2

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x

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y

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3units 15

512

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21

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4line about the

rotated is 4 when generated volume theFind 22

xyxii

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rotated is 4 when generated volume theFind 22

xyxii

x

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rotated is 4 when generated volume theFind 22

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4line about the

rotated is 4 when generated volume theFind 22

xyxii

x

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x

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4line about the

rotated is 4 when generated volume theFind 22

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rotated is 4 when generated volume theFind 22

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rotated is 4 when generated volume theFind 22

xyxii

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4line about the

rotated is 4 when generated volume theFind 22

xyxii

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x

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rotated is 4 when generated volume theFind 22

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x

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x

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rotated is 4 when generated volume theFind 22

xyxii

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x

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rotated is 4 when generated volume theFind 22

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x

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rotated is 4 when generated volume theFind 22

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x

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x

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242 x

4line about the

rotated is 4 when generated volume theFind 22

xyxii

x

422 yx

4

x

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x

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xxxA 4242 2

4line about the

rotated is 4 when generated volume theFind 22

xyxii

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x

y

2

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x

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4line about the

rotated is 4 when generated volume theFind 22

xyxii

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422 yx

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x

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x

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x42

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xxxA 4242 2

xxxV 2444

2444 xx

2

2

2

0444lim

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2

2

2

0444lim

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2

2

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2

2

2

0444lim

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2

2

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2

2

22

2

2 44444 dxxxdxx

2

2

2

0444lim

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2

2

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2

2

22

2

2 44444 dxxxdxx

2

2

22

2

2 44416 dxxxdxx

2

2

2

0444lim

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2

2

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2

2

22

2

2 44444 dxxxdxx

2

2

22

2

2 44416 dxxxdxx

semi-circle

2

2

2

0444lim

xxxxxV

2

2

2444 dxxx

222116

2

2

22

2

2 44444 dxxxdxx

2

2

22

2

2 44416 dxxxdxx

semi-circle

2

2

2

0444lim

xxxxxV

2

2

2444 dxxx

222116

2

2

22

2

2 44444 dxxxdxx

2

2

22

2

2 44416 dxxxdxx

semi-circle function oddeven odd

2

2

2

0444lim

xxxxxV

2

2

2444 dxxx

222116

2

2

22

2

2 44444 dxxxdxx

2

2

22

2

2 44416 dxxxdxx

semi-circle function oddeven odd

0

2

2

2

0444lim

xxxxxV

2

2

2444 dxxx

222116

2

2

22

2

2 44444 dxxxdxx

2

2

22

2

2 44416 dxxxdxx

semi-circle function oddeven odd

0

32units32

2

2

2

0444lim

xxxxxV

2

2

2444 dxxx

222116

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 oddeven odd

0

32units32