Volumes of Revolution

xy

0

We’ll first look at the area between the linesy = x , . . .

Ans: A cone ( lying on its side )

Can you see what shape you will get if you rotate the area through about the x-axis?360

x = 1, . . . and the x-axis. 1

Volumes of Revolution

xy

0 1

r

h

311,1 VhrFor this

cone,

hrV 231

We’ll first look at the area between the linesy = x , . . .

x = 1, . . . and the x-axis.

Volumes of Revolution

x

The formula for the volume found by rotating any area about the x-axis is

dxyVb

a 2

a and b are the x-coordinates at the left- and right-hand edges of the area.

where is the curve forming the upper edge of the area being rotated.

)(xfy

a b

)(xfy

We leave the answers in terms of

Volumes of Revolution

dxV 1

0 2x

r

h

0 1

So, for our cone, using integration, we get

1

0

3

3

x

31

03

1

We must substitute for y using before we integrate.

)(xfy

dxyVb

a 2

xy

I’ll outline the proof of the formula for you.

Volumes of Revolution

x

The formula can be proved by splitting the area into narrow strips. . .

Each tiny piece is approximately a cylinder ( think of a penny on its side ).

which are rotated about the x-axis.

Each piece, or element, has a volume hr 2 2y

y

Volumes of RevolutionThe formula can be proved by

splitting the area into narrow strips. . .

Each tiny piece is approximately a cylinder ( think of a penny on its side ).

x

Each piece, or element, has a volume hr 2 2y dx

dx

which are rotated about the x-axis. y

Volumes of RevolutionThe formula can be proved by

splitting the area into narrow strips. . .

Each tiny piece is approximately a cylinder ( think of a penny on its side ).

x

Each piece, or element, has a volume hr 2 2y dx

dx

dxyVb

a 2

The formula comes from adding an infinite number of these elements.

which are rotated about the x-axis. y

Volumes of Revolution

Solution: To find a volume we don’t need a sketch unless we are not sure what limits of integration we need. However, a sketch is often helpful.As these are the first examples I’ll sketch the curves.

xey

)1( xxy e.g. 1(a) The area formed by the curve and the x-axis from x = 0 to x = 1 is rotated through radians about the x-axis. Find the volume of the solid formed.

2

(b) The area formed by the curve , the x-axis and the lines x = 0 and x = 2 is rotated through radians about the x-axis. Find the volume of the solid formed.

2

Volumes of Revolution

)1( xxy

dxyVb

a 2

area rotate about the x-axis

A common error in finding a volume is to get wrong. So beware!

2y

)1( xxy 222 )1( xxy

)21( 222 xxxy 4322 2 xxxy

(a) rotate the area between.10)1( tofrom axis- the and xxxy

Volumes of Revolution

)1( xxy

dxyVb

a 24322 2 xxxy

dxxxxV 1

0

432 2a = 0, b = 1

(a) rotate the area between.10)1( tofrom axis- the and xxxy

Volumes of Revolution

dxxxxV 1

0

432 21

0

543

54

2

3

xxx

0

5

1

2

1

3

1

30

1

30

2

Volumes of Revolution

xey

(b) Rotate the area between xey and the lines x = 0 and x = 2.

dxyVb

a 2 22 xey

xxx eee2 xe 2

2x

Volumes of Revolution

xey

(b) Rotate the area between xey and the lines x = 0 and x = 2.

dxyVb

a 2dxeV x

2

0

2

22 xey xe 2

2x

Volumes of Revolution

dxeV x2

0

22

0

2

2

xe

22

04 ee Remember that 0e 1

2

1

2

4e

2

14e

Volumes of RevolutionExercis

e

xy

1

radians about the x-axis. Find the volume of the solid formed.

2

1(a) The area formed by the curve the x-

axis and the lines x = 1 to x = 2 is rotated

through

(b) The area formed by the curve , the x-axis and the lines x = 0 and x = 2 is rotated through radians about the x-axis. Find the volume of the solid formed.

2

xxy

Volumes of RevolutionSolutions

:1. x

y1

(a), the x-axis and the lines x = 1 and x = 2.

dxyVb

a 2 dxx

V 2

12

1

dxxV 2

1

22

1

1

1

x

V

2

1

1

xV

Volumes of RevolutionSolutions

: 2

1

1

xV

1

2

1V

2

V

Volumes of Revolution

Solution:

(b) xxy , the x-axis and the lines x = 0 and x = 2.

dxyVb

a 2

xxy 32 xxxxxy

dxxV 2

0

32

0

4

4

xV

4 V

Volumes of Revolution

To rotate an area about the y-axis we use the same formula but with x and y swapped.

dxyVb

a 2 dyxVd

c 2

The limits of integration are now values of y giving the top and bottom of the area that is rotated.

Rotation about the y-axis

As we have to substitute for x from the equation of the curve we will have to rearrange the equation.

Tip: dx for rotating about the x-axis;

dy for rotating about the y-axis.

Volumes of Revolution

xy

dyxVd

c 2

e.g. The area bounded by the curve , the y-axis and the line y = 2 is rotated through about the y-axis. Find the volume of the solid formed.

xy 360

2y

dyyV 2

0

4

xy 2yx 42 yx

Volumes of Revolution

dyyV 2

0

42

0

5

5

y

0

5

52

5

32

Volumes of RevolutionExercis

e2xy 0x

the y-axis and the line y = 3 is rotated through radians about the y-axis. Find the volume of the solid formed.

2

1(a) The area formed by the curve for

xy

1(b) The area formed by the curve ,

the y-axis and the lines y = 1 and y = 2 is

rotated 2through radians about the y-axis. Find the volume of the solid formed.

Volumes of Revolution

2xy

Solutions: 2xy (a) for , the y-axis and the line y =

3.0x

3

0dyyV

dyxVd

c 2

3

0

2

2

y2

9

Volumes of Revolution

Solution:

2

1 2

1dy

yV

dyxVd

c 2

2

1

1

y

1

2

1

(b)x

y1

, the y-axis and the lines y = 1 and y =

2.

2

yx

xy

11

22 1

yx

Volumes of Revolution