56
rotati || slice Volumes By Cylindrical Shells

X2 T06 02 Cylindrical Shells

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Page 1: X2 T06 02 Cylindrical Shells

rotation || slice

Volumes By Cylindrical Shells

Page 2: X2 T06 02 Cylindrical Shells

rotation || slice

Volumes By Cylindrical Shells

y

x

xfy

Page 3: X2 T06 02 Cylindrical Shells

rotation || slice

Volumes By Cylindrical Shells

y

x

xfy

a b

Page 4: X2 T06 02 Cylindrical Shells

rotation || slice

Volumes By Cylindrical Shells

y

x

xfy

a x b

Page 5: X2 T06 02 Cylindrical Shells

rotation || slice

Volumes By Cylindrical Shells

y

x

xfy

a x b

Page 6: X2 T06 02 Cylindrical Shells

rotation || slice

Volumes By Cylindrical Shells

y

x

xfy

a x b

Page 7: X2 T06 02 Cylindrical Shells

rotation || slice

Volumes By Cylindrical Shells

y

x

xfy

a xx

b

Page 8: X2 T06 02 Cylindrical Shells

rotation || slice

Volumes By Cylindrical Shells

y

x

xfy

a x

y

xb

Page 9: X2 T06 02 Cylindrical Shells

rotation || slice

Volumes By Cylindrical Shells

y

x

xfy

a x

y

x

xf

b

Page 10: X2 T06 02 Cylindrical Shells

rotation || slice

Volumes By Cylindrical Shells

y

x

xfy

a x

y

x

xf

x2b

Page 11: X2 T06 02 Cylindrical Shells

rotation || slice

Volumes By Cylindrical Shells

y

x

xfy

a x

y

x

xf

xfxxA 2

x2b

Page 12: X2 T06 02 Cylindrical Shells

rotation || slice

Volumes By Cylindrical Shells

y

x

xfy

a x

y

x

xf

xfxxA 2

xxfxV 2

x2b

Page 13: X2 T06 02 Cylindrical Shells

rotation || slice

Volumes By Cylindrical Shells

y

x

xfy

a x

y

x

xf

xfxxA 2

xxfxV 2

b

axx

xxfxV 2lim0

x2b

Page 14: X2 T06 02 Cylindrical Shells

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

Page 15: X2 T06 02 Cylindrical Shells

axis about the rotated is axis the

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

xx

xyi

Page 16: X2 T06 02 Cylindrical Shells

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

Page 17: X2 T06 02 Cylindrical Shells

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

Page 18: X2 T06 02 Cylindrical Shells

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

Page 19: X2 T06 02 Cylindrical Shells

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

Page 20: X2 T06 02 Cylindrical Shells

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

Page 21: X2 T06 02 Cylindrical Shells

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

Page 22: X2 T06 02 Cylindrical Shells

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

Page 23: X2 T06 02 Cylindrical Shells

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

Page 24: X2 T06 02 Cylindrical Shells

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

Page 25: X2 T06 02 Cylindrical Shells

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

Page 26: X2 T06 02 Cylindrical Shells

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

Page 27: X2 T06 02 Cylindrical Shells

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

Page 28: X2 T06 02 Cylindrical Shells

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

Page 29: X2 T06 02 Cylindrical Shells

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

Page 30: X2 T06 02 Cylindrical Shells

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

Page 31: X2 T06 02 Cylindrical Shells

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

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4

2

1

4 yx

2

1

42 y

y2

4

0

2

1

44 dyyy

4

0

2

3

2

1

44 dyyy

Page 32: X2 T06 02 Cylindrical Shells

4 line about the

rotated is 4 when generated volume theFind 22

x

yxii

Page 33: X2 T06 02 Cylindrical Shells

4 line about the

rotated is 4 when generated volume theFind 22

x

yxii

x

422 yx

y

2

-2

2-2

Page 34: X2 T06 02 Cylindrical Shells

4 line about the

rotated is 4 when generated volume theFind 22

x

yxii

x

422 yx

4

y

2

-2

2-2

Page 35: X2 T06 02 Cylindrical Shells

4 line about the

rotated is 4 when generated volume theFind 22

x

yxii

x

422 yx

4

x

y

2

-2

2-2

Page 36: X2 T06 02 Cylindrical Shells

4 line about the

rotated is 4 when generated volume theFind 22

x

yxii

x

422 yx

4

x

y

2

-2

2-2

Page 37: X2 T06 02 Cylindrical Shells

4 line about the

rotated is 4 when generated volume theFind 22

x

yxii

x

422 yx

4

x

y

2

-2

2-2

Page 38: X2 T06 02 Cylindrical Shells

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

Page 39: X2 T06 02 Cylindrical Shells

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

Page 40: X2 T06 02 Cylindrical Shells

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

Page 41: X2 T06 02 Cylindrical Shells

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

Page 42: X2 T06 02 Cylindrical Shells

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

Page 43: X2 T06 02 Cylindrical Shells

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

Page 44: X2 T06 02 Cylindrical Shells

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

Page 45: X2 T06 02 Cylindrical Shells

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

Page 46: X2 T06 02 Cylindrical Shells

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

Page 47: X2 T06 02 Cylindrical Shells

2

2

2

0444lim

xx

xxxV

Page 48: X2 T06 02 Cylindrical Shells

2

2

2

0444lim

xx

xxxV

2

2

2444 dxxx

Page 49: X2 T06 02 Cylindrical Shells

2

2

2

0444lim

xx

xxxV

2

2

2444 dxxx

2

2

22

2

2 44444 dxxxdxx

Page 50: X2 T06 02 Cylindrical Shells

2

2

2

0444lim

xx

xxxV

2

2

2444 dxxx

2

2

22

2

2 44444 dxxxdxx

2

2

22

2

2 44416 dxxxdxx

Page 51: X2 T06 02 Cylindrical Shells

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

Page 52: X2 T06 02 Cylindrical Shells

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

Page 53: X2 T06 02 Cylindrical Shells

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

Page 54: X2 T06 02 Cylindrical Shells

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

Page 55: X2 T06 02 Cylindrical Shells

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

Page 56: X2 T06 02 Cylindrical Shells

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