X2 t05 02 cylindrical shells (2012)

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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

24 xy

-2 2

y4

axisabout therotatedis axis the

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

xxxyi

x

24 xy

-2 2

y

y4

axisabout therotatedis axis the

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

xxxyi

x

24 xy

-2 2

y

y4

axisabout therotatedis axis the

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

xxxyi

x

24 xy

-2 2

y

y4

y

axisabout therotatedis axis the

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

xxxyi

x

24 xy

-2 2

y

y4

y

x2

21

4 yx

axisabout therotatedis axis the

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

xxxyi

x

24 xy

-2 2

y

y4

y

x2

21

4 yx

axisabout therotatedis axis the

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

xxxyi

x

24 xy

-2 2

y

y4

y

x2

21

4 yx

21

42 y

axisabout therotatedis axis the

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

xxxyi

x

24 xy

-2 2

y

y4

y

x2

21

4 yx

21

42 y

y2

axisabout therotatedis axis the

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

xxxyi

x

24 xy

-2 2

y

y4

y

x2

2

1422 yyyA

21

4 yx

21

42 y

y2

axisabout therotatedis axis the

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

xxxyi

x

24 xy

-2 2

y

y4

y

x2

2

1422 yyyA

yyyV 21

44

21

4 yx

21

42 y

y2

axisabout therotatedis axis the

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

xxxyi

x

24 xy

-2 2

y

y4

y

x2

2

1422 yyyA

yyyV 21

44

4

0

21

044lim

xyyyyV

21

4 yx

21

42 y

y2

axisabout therotatedis axis the

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

xxxyi

x

24 xy

-2 2

y

y4

y

x2

2

1422 yyyA

yyyV 21

44

4

0

21

044lim

xyyyyV

4

0

21

44 dyyy 21

4 yx

21

42 y

y2

axisabout therotatedis axis the

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

xxxyi

x

24 xy

-2 2

y

y4

y

x2

2

1422 yyyA

yyyV 21

44

4

0

21

044lim

xyyyyV

4

0

21

44 dyyy 21

4 yx

21

42 y

y2

4

0

21

44 dyyy

axisabout therotatedis axis the

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

xxxyi

x

24 xy

-2 2

y

y4

y

x2

2

1422 yyyA

yyyV 21

44

4

0

21

044lim

xyyyyV

4

0

21

44 dyyy 21

4 yx

21

42 y

y2

4

0

21

44 dyyy

4

0

23

21

44 dyyy

axisabout therotatedis axis the

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

xxxyi

x

24 xy

-2 2

y

y4

y

x2

2

1422 yyyA

yyyV 21

44

4

0

21

044lim

xyyyyV

4

0

21

44 dyyy

4

0

25

23

52

384

yy

21

4 yx

21

42 y

y2

4

0

21

44 dyyy

4

0

23

21

44 dyyy

axisabout therotatedis axis the

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

xxxyi

x

24 xy

-2 2

y

y4

y

x2

2

1422 yyyA

yyyV 21

44

4

0

21

044lim

xyyyyV

4

0

21

44 dyyy

4

0

25

23

52

384

yy

3units 15

512

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21

4 yx

21

42 y

y2

4

0

21

44 dyyy

4

0

23

21

44 dyyy

4line about the

rotated is 4 when generated volume theFind 22

xyxii

4line about the

rotated is 4 when generated volume theFind 22

xyxii

x

422 yx

y

2

-2

2-2

4line about the

rotated is 4 when generated volume theFind 22

xyxii

x

422 yx

4

y

2

-2

2-2

4line about the

rotated is 4 when generated volume theFind 22

xyxii

x

422 yx

4

x

y

2

-2

2-2

4line about the

rotated is 4 when generated volume theFind 22

xyxii

x

422 yx

4

x

y

2

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2-2

4line about the

rotated is 4 when generated volume theFind 22

xyxii

x

422 yx

4

x

y

2

-2

2-2

4line about the

rotated is 4 when generated volume theFind 22

xyxii

x

422 yx

4

x

y

2

-2

2-2

torus

4line about the

rotated is 4 when generated volume theFind 22

xyxii

x

422 yx

4

x

y

2

-2

2-2

torus

4line about the

rotated is 4 when generated volume theFind 22

xyxii

x

422 yx

4

x

y

2

-2

2-2

torus

x

4line about the

rotated is 4 when generated volume theFind 22

xyxii

x

422 yx

4

x

y

2

-2

2-2

torus

x

y2

4line about the

rotated is 4 when generated volume theFind 22

xyxii

x

422 yx

4

x

y

2

-2

2-2

torus

x

y2242 x

4line about the

rotated is 4 when generated volume theFind 22

xyxii

x

422 yx

4

x

y

2

-2

2-2

torus

x

y2

x42

242 x

4line about the

rotated is 4 when generated volume theFind 22

xyxii

x

422 yx

4

x

y

2

-2

2-2

torus

x

y2

x42

242 x

xxxA 4242 2

4line about the

rotated is 4 when generated volume theFind 22

xyxii

x

422 yx

4

x

y

2

-2

2-2

torus

x

y2

x42

242 x

xxxA 4242 2

2444 xx

4line about the

rotated is 4 when generated volume theFind 22

xyxii

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

xxxxxV

2

2

2

0444lim

xxxxxV

2

2

2444 dxxx

2

2

2

0444lim

xxxxxV

2

2

2444 dxxx

2

2

22

2

2 44444 dxxxdxx

2

2

2

0444lim

xxxxxV

2

2

2444 dxxx

2

2

22

2

2 44444 dxxxdxx

2

2

22

2

2 44416 dxxxdxx

2

2

2

0444lim

xxxxxV

2

2

2444 dxxx

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