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