What is the grey area ? Take strip perpendicular to x-axis What are the limits of integration.

What is the grey area ?

Take strip perpendicular to x-axis

What are the limits of integration.

(1 ) 02

xA dx

2

0(1 )

2

xA dx

What is the grey area ?

Take strip parallel to x-axis

What are the limits of integration.

(2 2 ) 0A y dy

1

0(2 2 ) 0A y dy

What is the volumeif the grey area is revolved about the

x-axis?

What are the limits of integration.

What is the volumeif the grey area is revolved about the

x-axis?

What are the limits of integration.

2 2

0(1 )

2

xV dx

What is the volumeif the yellow area is revolved about the

y-axis?

What is the volumeif the yellow area is revolved about the

y-axis?2 2

0

3( )2

yV dy

When finding the area, use

A. True

B. False

2

0( ) ( )top x bottom x dx

When finding the area, use

A. True

B. False

2

0( ) ( )top x bottom x dx

When finding the area, use

A. True

B. False

3

0

2(2 )

3x dx

When finding the area, use

A. True

B. False

3

0

2(2 )

3x dx

Revolve about the y-axisV =

A. True

B. False

3

0

22 (2 )

3x x dx

Revolve about the y-axisV =

A. True

B. False

3

0

22 (2 )

3x x dx

Revolve about the x-axisV =

A. True

B. False

3 2

0

4(4 )

9x dx

Revolve about the x-axisV =

A. True

B. False

3 2

0

4(4 )

9x dx

What is the volumeif the grey area is revolved about the

x-axis?Red strip perpendicular to axis

Solve for y

and square

12

xy

What is the volumeif the grey area is revolved about the

x-axis?

What are the limits of integration.

2 2

0(1 )

2

xV dx

What is the volumeif the yellow area is revolved about the

y-axis?

Solve for x, square, integrate, times pi

What is the volumeif the yellow area is revolved about the

y-axis?

Solve for x, square, integrate, times pi

2 2

0

3( )2

yV dy

#56 Plumb bob design

Revolve the shown region about the x-axis

23612

xy x

#56 Plumb bob design

Revolve the shown region about the x-axis

Must weigh in the neighborhood of 190 g.

Specify a brass that weighs 8.5 g/cm3.

23612

xy x

#56 Plumb bob design

V =

Must weigh in the neighborhood of 190 g.

Specify a brass that weighs 8.5 g/cm3.

23612

xy x

22(36 )

144

xx dx

#56 Plumb bob design

V =

=22.62

cm3

times 8.5 g/cm3.

=192.27 g.

23612

xy x

26 2

0(36 )

144

xx dx

6 2 4

036

144x x dx

3 5 56 3

0

36 6( ) (12(6 ) )

144 3 5 144 5|x x

.2 2

0 02 sin cos sin 2V x x dx x dx

.

3.14159

0.1

2 2

0 02 sin cos sin 2V x x dx x dx

What is the volumeif the yellow area is rotated about the

y-axis?

Last time we went about the x as shown.

Dr. Jack Tenzel has a Project-o-Chart in his office

• The light reflects off of a mirror and ends up on a wall in front of the patient.

Given the center light source, calculate the volume around it

• First write the equation of the surface.

y = 16 x-4

We will come

back to this later.

What is the volume of a coke can?

• Just the aluminum

• Top - Bottom

The volume of a can is 2 r times the height times

the x.

Start like we did for area. Take a narrow x red strip and then

rotate it about the y-axis.

• This makes a red coke can.• The volume of one can is…• 2x(f(x)-g(x))x so the desired volume is

Set up n rectangles of width x

Revolve about the y-axis

That produces n cylinders

* *

1

lim 2 ( ( ) ( ) i

n

i in

i

Volume x f x x xg

Take a narrow x red strip and then rotate it about the y-axis.

• This makes a coke can.

• The volume of one can with radius x is…

• 2x(f(x)-g(x))x so the desired volume is

* *

1

lim 2 ( ( ) ( ) i

n

i in

i

Volume x f x x xg

By the definition

• Volume = * *

1

lim 2 ( ( ) ( )n

i i in

i

x f x g x x

2 ( ( ) ( ))

b

a

x f x g x dx

Example 1

• Find the volume when the area under y = x2 and over the x-axis is revolved about the y-axis.

• Between x=0 and x=2• Just add up all of the red coke cans• As they slide from x=0 to x=2• Top function is y= x2

• Bottom function is y = 0

Example 1• Find the volume when the area under

y=x2

• Between x=0 and x=2• Is revolved about the y-axis

• = 2x4/4

• = 2/4 = 8

22

0

2 ( 0)Volume x x dx 20|

.

• Back to the problem

• x is the radius times top - bottom

2 ( ( ) ( ))b

a

V x f x g x dx

24

1

2 (16 1)V x x dx

.2

4

1

2 (16 1)V x x dx 2

3

1

2 (16 )x x dx 2 22 2 2

21 1

| 8 |2 (16 ) 2 ( )

2 2 | 2 |

x x x

x

A. 2B. 2C. 2

22

21

8 |2 ( )

2 |

x

x

A. 2B. 2C. 2

22

21

8 |2 ( )

2 |

x

x

2

A. 2[ - 4 + 8.5]

B. 2[ - 4 - 8.5]

C. 2[ - 4 + 7.5 ]

2

A. 2[ - 4 + 8.5]

B. 2[ - 4 - 8.5]

C. 2[ - 4 + 7.5 ]

Example 3

• Revolve the area between x2 and x3 about the y-axis• Find the volume generated.• 0 = x2 ( x – 1 ) • so x2 =0 or x–1=0• Next we add up all of the red cylinders• From 0 to 1• Volume =

Revolve about the y-axisA. [

B. [

C. [

12 3

0

. 2 [ ]A V x x x dx 2

2 3

0

. 2 [ ]B V x x x dx 1

3 2

0

. 2 [ ]C V x x x dx

Revolve about the y-axisA. [

B. [

C. [

12 3

0

. 2 [ ]A V x x x dx 2

2 3

0

. 2 [ ]B V x x x dx 1

3 2

0

. 2 [ ]C V x x x dx

.

A. .

B. .

C. .

12 3 4

0

|. 2 ( [ ])

2 3 4 |

x x xA

14 5

0

|. 2 [ ]

4 5 |

x xB V

12 3

0

|. 2 [3 4 ]

|C V x x

12 3

0

2 [ ]V x x x dx

.

A. .

B. .

C. .

12 3 4

0

|. 2 ( [ ])

2 3 4 |

x x xA

14 5

0

|. 2 [ ]

4 5 |

x xB V

12 3

0

|. 2 [3 4 ]

|C V x x

12 3

0

2 [ ]V x x x dx

Volume =

= 2 [(5-4)/20] = 2 /20 = /10

12 3

0

2 ( )V x x x dx

4 510

1 12 [ ] | 2 [ ]

4 5 4 5

x x

Example 2

• Consider the first region in the first quadrant bounded by y=sin(x2) and y=1/root(2)

• Set the two functions equal• Solve for x2 and then for x• Spin about the y-axis or radius of x• Add the volumes of the n cylinders

* *2

1

22 lim (sin( ) )

2

n

i i ini

V x x x

Example 2

• sin(x2) = root(2)/2 when x2=/4 or 3/4

* *2

1

22 lim (sin( ) )

2

n

i i ini

V x x x

3

22

2

22 (sin( ) )

2V x x dx

Example 2

3

22

2

22 (sin( ) )

2V x x dx

3

22

2

(sin( )2 2 )V x x x dx

Example 22 3

2 2

2

[ cos( ) 2 ] |2

xx

3

22

2

sin( )2 2V x x x dx

3 3 2 2cos cos

4 8 4 8

2 3 2 2 2

2 8 2 8

2

24

y = x + 3 .…….y = x2 + 1 . …1

• 2 pi radius ( top – bottom) thick.

2 2

12 ( 1)([ 3] [ 1])V x x x dx

.

A. .

B. .

C. .

2 2

12 ( 1)([ 3] [ 1])V x x x dx

2 3

12 3 2x x dx

2 3

12 3 4x x dx

2 3 2

12 3 4x x dx

.

A. .

B. .

C. .

2 2

12 ( 1)([ 3] [ 1])V x x x dx

2 3

12 3 2x x dx

2 3

12 3 4x x dx

2 3 2

12 3 4x x dx

Region of y=x2+1 and y=x+3 is revolved about the line x = -1.

2 2

12 ( 1)([ 3] [ 1])V x x x dx

2 3

12 3 2x x dx

Region of y=x2+1 and y=x+3 is revolved about the line x = -1.

45.95

0.2

2 2

12 ( 1)([ 3] [ 1])V x x x dx

2 3

12 3 2x x dx

Region bounded by y=x2+1 and y=x+3 is revolved about the x-axis.

Region bounded by y=x2+1 and y=x+3 is revolved about the x-axis.

73.51

0.2