Volumes of Solids of Revolution

y

x

y = f(x)

Volumes of Solids of Revolution

y

x

y = f(x)

a b

Volumes of Solids of Revolution

y

x

y = f(x)

a b

Volumes of Solids of Revolution

y

x

y = f(x)

a b

Volume of a solid of revolution about;

Volumes of Solids of Revolution

y

x

y = f(x)

a b

b

adxyVxi 2:axis Volume of a solid of revolution about;

Volumes of Solids of Revolution

y

x

y = f(x)

a b

b

adxyVxi 2:axis Volume of a solid of revolution about;

d

cdyxVyii 2:axis

e.g. (i) cone

y

x

mxy

h

e.g. (i) cone

y

x

mxy

h

e.g. (i) cone

y

x

mxy

h

dxxm

dxyV

0

22

2

h

e.g. (i) cone

y

x

mxy

h

dxxm

dxyV

0

22

2

h

xm0

32

31

h

e.g. (i) cone

y

x

mxy

h

dxxm

dxyV

0

22

2

h

xm0

32

31

32

32

31

031

hm

hm

h

e.g. (i) cone

y

x

mxy

h

dxxm

dxyV

0

22

2

h

xm0

32

31

32

32

31

031

hm

hm

h

r

e.g. (i) cone

y

x

mxy

h

dxxm

dxyV

0

22

2

h

xm0

32

31

32

32

31

031

hm

hm

h

r

hrm

e.g. (i) cone

y

x

mxy

h

dxxm

dxyV

0

22

2

h

xm0

32

31

32

32

31

031

hm

hm

h

r

hrm

3

3

31

2

2

32

32

hrhhr

hhr

(ii) sphere

y

x

222 ryx

r

(ii) sphere

y

x

222 ryx

r

(ii) sphere

y

x

222 ryx

r

r

dxxr

dxyV

0

22

2

2

(ii) sphere

y

x

222 ryx

r

r

dxxr

dxyV

0

22

2

2

r

xxr0

32

312

(ii) sphere

y

x

222 ryx

r

r

dxxr

dxyV

0

22

2

2

r

xxr0

32

312

3

3

32

34

322

0312

r

r

rrr

(iii) .1 and 0between axis thearound

revolved is when generated solid theof volume theFind 2

yyyxy

(iii) .1 and 0between axis thearound

revolved is when generated solid theof volume theFind 2

yyyxy

y

x

2xy

(iii) .1 and 0between axis thearound

revolved is when generated solid theof volume theFind 2

yyyxy

y

x

2xy

1

(iii) .1 and 0between axis thearound

revolved is when generated solid theof volume theFind 2

yyyxy

y

x

2xy

1

(iii) .1 and 0between axis thearound

revolved is when generated solid theof volume theFind 2

yyyxy

y

x

2xy

1

1

0

2

ydy

dyxV

(iii) .1 and 0between axis thearound

revolved is when generated solid theof volume theFind 2

yyyxy

y

x

2xy

1

1

0

2

ydy

dyxV

102

2y

(iii) .1 and 0between axis thearound

revolved is when generated solid theof volume theFind 2

yyyxy

y

x

2xy

1

1

0

2

ydy

dyxV

102

2y

3

2

units 2

012

(iv)

y

x

xy 5

Find the volume of the solid when the shaded region is rotated about the x axis

5

(iv)

y

x

xy 5

1

0

2

1

0

22

2

2525

55

dxx

dxx

dxyV

Find the volume of the solid when the shaded region is rotated about the x axis

5

(iv)

y

x

xy 5

1

0

2

1

0

22

2

2525

55

dxx

dxx

dxyV

1

0

3

3125

xx

Find the volume of the solid when the shaded region is rotated about the x axis

5

(iv)

y

x

xy 5

1

0

2

1

0

22

2

2525

55

dxx

dxx

dxyV

1

0

3

3125

xx

3

3

unit 3

50

0131125

Find the volume of the solid when the shaded region is rotated about the x axis

5

(iv)

y

x

xy 5

1

0

2

1

0

22

2

2525

55

dxx

dxx

dxyV

1

0

3

3125

xx

3

3

unit 3

50

0131125

Find the volume of the solid when the shaded region is rotated about the x axis

5

OR

(iv)

y

x

xy 5

1

0

2

1

0

22

2

2525

55

dxx

dxx

dxyV

1

0

3

3125

xx

3

3

unit 3

50

0131125

Find the volume of the solid when the shaded region is rotated about the x axis

5

OR

3

2

22

units 3

50

1532

31

hrhrV

2005 HSC Question 6c)

The graphs of the curves and are shown in the diagram.

2y x 212 2y x

(i) Find the points of intersection of the two curves. (1)

2005 HSC Question 6c)

The graphs of the curves and are shown in the diagram.

2y x 212 2y x

(i) Find the points of intersection of the two curves. (1)2 212 2x x

2005 HSC Question 6c)

The graphs of the curves and are shown in the diagram.

2y x 212 2y x

(i) Find the points of intersection of the two curves. (1)2 212 2x x 2

2

3 124

2

xxx

2005 HSC Question 6c)

The graphs of the curves and are shown in the diagram.

2y x 212 2y x

(i) Find the points of intersection of the two curves. (1)2 212 2x x 2

2

3 124

2

xxx

meet at 2,4

(ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into twoparts, or otherwise, find the volume of the solid formed.

(ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into twoparts, or otherwise, find the volume of the solid formed.

(ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into twoparts, or otherwise, find the volume of the solid formed.

2V x dy

(ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into twoparts, or otherwise, find the volume of the solid formed.

2V x dy

(ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into twoparts, or otherwise, find the volume of the solid formed.

2V x dy

2x y

(ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into twoparts, or otherwise, find the volume of the solid formed.

2V x dy

2x y

2 162

x y

(ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into twoparts, or otherwise, find the volume of the solid formed.

2V x dy

2x y

2 162

x y

4 12

0 4

162

V y dy ydy

(ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into twoparts, or otherwise, find the volume of the solid formed.

2V x dy

2x y

2 162

x y

4 12

0 4

162

V y dy ydy

4 122 2

0 4

1 164 2

y y y

(ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into twoparts, or otherwise, find the volume of the solid formed.

2V x dy

2x y

2 162

x y

4 12

0 4

162

V y dy ydy

4 122 2

0 4

1 164 2

y y y

2 2 216 4 4 0 12 44 2

(ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into twoparts, or otherwise, find the volume of the solid formed.

2V x dy

2x y

2 162

x y

4 12

0 4

162

V y dy ydy

4 122 2

0 4

1 164 2

y y y

2 2 216 4 4 0 12 44 2

384 units

Exercise 11G; 3bdef, 4eg, 5cd, 6d, 8ad, 12, 16a, 19 to 22, 24*, 25*