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A Note For You

In the disk method, we add up the volumes of an infinite number of infinitesimally thin circular disks to find the total volume of a solid.

2

2

2

π

π ( )

π ( )

onedisk

b

a

V r h

f x dx

V f x dx

=

=

=

b

Rr d

c

f(y)g(y)

Revolved about the ___-axis:

( )( )

( )

2 2

2 2

2 2

2 2

π π

π

π ( ) ( )

π ( ) ( )

onedisk

b

a

V R h r h

R r h

f x g x dx

V f x g x dx

= −

= −

= −

= −

( )( )

( )

2 2

2 2

2 2

2 2

π π

π

π ( ) ( )

π ( ) ( )

onedisk

d

c

V R h r h

R r h

f y g y h

V f y g y dy

= −

= −

= −

= −

2

2

2

π

π ( )

π ( )

onedisk

d

c

V r h

g y dy

V g y dy

=

=

=

g(y)

Volume of each Shell = 2prh*thickness of the shellNote: this is NOT the volume inside the shell or cylinder, but the volume of the cylinder itself, like each onion ring.

In the cylindrical shell method, we add up the volumes of an infinite number of infinitesimally thin cylinders to find the total volume of a solid (similar to a an onion).

2π ( )

2π ( ) ( )

oneshell

a

b

thicknessof the shellV rh

V r x h x dx

= ⋅

=

Revolved about the __ -axis:

©2018 joan kessler distancemath.com

Revolved about the __-axis:

Revolved about the ___-axis:

Name_______________

Revolved about the ___-axis:

2

10yx

= Bounds: y = 0, x = 1, x = 5

Find the exact volume of the boundedregion revolved about the x-axis using the disk method. Sketch & label.

1 2

2

10yx

= Bounds: y = 0, x = 1, x = 5

Find the exact volume of the bounded region revolved about the y-axis using the shell method. Sketch and label.

34

2πb

aV r dx=

2

10yx

= y = 0, x = 1, x = 5

Find the exact volume of the bounded region revolved about the line y= 10 using the washer method. Sketch and label..

Bounds:

Find the exact volume of the bounded by the graphs of y = 4x - x2 and y = x2 about the line x = 2. Use the Shell method. Sketch and label.

©2018 joan kessler distancemath.com

Name_______________

2πb

aV rhdx=

5

©2018 joan kessler distancemath.com

Find the exact volume of the solid of

revolution formed by rotating the region

bounded by the x-axis and the graph of

about the x-axis. Sketch.

3 32 cos(3 ) 0 2π .y x x x x= − = =from to

For Calculator use window [0,2] x [-1,4]

Find the exact volume of the solid of revolution

formed by rotating the region bounded by the graphs of

about the x-axis on [0, 5]. Sketch.

65 sin(2 ) 2 cos(2 ).y x and y xπ π= + = +

___________V =

Which method is this?______

Find the exact volume of the solid

of revolution formed by rotating the

region bounded the graphs of

7

2( ) 3 ( )f x x g x x= =and about the y - axis

Sketch and label

What is g(y), the inner radius?________________

What is f(y), the outer raius?________________

Use the washer method.

What is the formula?______

What are points of intersection?

What are the limits of integration? y = ____ Y = ____

Name_______________

Suppose you have to determine the exact volume of

the solid of revolution by revolving the area bounded by

the parabola x = 4 - 3y - y2 and

x = 0 and the x-axis about the x-axis.

It appears that the disk method would be the easiest

method, but when looking at

the diagram and trying to

sub into the formula

we realize that the parabola is not a function and it would

be very difficult to solve for y= f(x). Try the Shell method.

8

r

2π ( )b

aV f x dx =

02π ( ) ( )a

around yV r x h x dx= Since the shell is revolved

around the x- axis we must

2π ( ) ( )d

around x cV r y h y dy= adjust the formula.

r

The radius r(y) the height of the curve, y, and h(y) is thelength of the cylinder. Find the volume.

h(y)

1 2

02π (4 - 3 - )around x y

V y y y dy=

=

r(y)

2π ( ) ( )d

cV r y h y dy=

Horizontal Axis of Rotation

( )h y

c

d

( )r y

2π ( ) ( )b

aV r x h x dy=

Vertical Axis of Rotation

b

( )h x

a ( )r x

( )h y

c

d

( )r y

b

( )h x

a ( )r x

Find the exact volume of the solid ofrevolution formed by rotating the region bounded by the graph of x = 9 - y2 and the y- axis on (0 < y < 3) about the x – axis. Sketch

Horizontal

Axis of Rotation

Vertical

Axis of Rotation

©2018 joan kessler distancemath.com

Name_______________

In the disk method, we add up the volumes of an infinite number of infinitesimally thin circular disks to find the total volume of a solid.

2

2

2

π

π ( )

π ( )

onedisk

b

a

V r h

f x dx

V f x dx

=

=

=

b

Rr d

c

f(y)g(y)

Revolved about the _x__-axis:

( )( )

( )

2 2

2 2

2 2

2 2

π π

π

π ( ) ( )

π ( ) ( )

onedisk

b

a

V R h r h

R r h

f x g x dx

V f x g x dx

= −

= −

= −

= −

( )( )

( )

2 2

2 2

2 2

2 2

π π

π

π ( ) ( )

π ( ) ( )

onedisk

d

c

V R h r h

R r h

f y g y h

V f y g y dy

= −

= −

= −

= −

2

2

2

π

π ( )

π ( )

onedisk

d

c

V r h

g y dy

V g y dy

=

=

=

g(y)

Volume of each Shell = 2prh*thickness of the shellNote: this is NOT the volume inside the shell or cylinder, but the volume of the cylinder itself, like each onion ring.

In the cylindrical shell method, we add up the volumes of an infinite number of infinitesimally thin cylinders to find the total volume of a solid (similar to a an onion).

2π ( )

2π ( ) ( )

oneshell

a

b

thicknessof the shellV rh

V r x h x dx

= ⋅

=

Revolved about the y -axis:

©2018 joan kessler distancemath.com

Revolved about the _y_-axis:

Revolved about the ___-axis:

Name_______________

Revolved about the ___-axis:y

x

2

10yx

= Bounds: y = 0, x = 1, x = 5

Find the exact volume of the boundedregion revolved about the x-axis using the disk method. Sketch & label.

1 2

2

10yx

= Bounds: y = 0, x = 1, x = 5

Find the exact volume of the bounded region revolved about the y-axis using the shell method. Sketch and label.

34

25

21

5 4

153

2

1

10

100

1003

496 103.8

π

815

b

a

V dxx

x dx

x

V r dx

π

π

π

π

−

−

=

=

= −

=

= ≈

( )

5

21

5

1

5

1

102

20

20 ln

20 ln5 ln120 ln5 101.124

2πb

a

V x dxxdxx

V rh

x

dx

π

π

π

ππ

=

=

=

=

= −= ≈

2

10yx

= y = 0, x = 1, x = 5

Find the exact volume of the bounded region revolved about the line y= 10 using the washer method. Sketch and label..

( )

( )

25 221

5

2 41

5 2 4

15

3

2 2

1

1010 10

200 100100 100

200 100

200 100 1904ππ 398.913

π

5

b

aV R r

V dxx

dxx x

x d

d

x x

x

x x

π

π

π − −

= − −

= − + −

= −

− = + = ≈

=

−

Bounds:

Find the exact volume of the bounded by the graphs of y = 4x - x2 and y = x2 about the line x = 2. Use the Shell method .Sketch and label..

©2018 joan kessler distancemath.com

( )( )

( )( )

2 2 2

0

2 2

02 3 2

024 3 2

0

2 2 4

2 2 4 2

2 2 8 8

2 8 82π 4 3 2

16π

2π ( )

16.76

(

3

)b

a

V x x x x dx

x x x dx

x x x dx

x x x

V r x h x dx

π

π

π

= − − −

= − −

= − +

= − +

=

= ≈

r h

2

2

4 22 0

( 2) 00,2

x x xx xx xx

− =

− =− =

−2

Name_______________

5

©2018 joan kessler distancemath.com

Find the exact volume of the solid of

revolution formed by rotating the region

bounded by the x-axis and the graph of

about the x-axis. Sketch.

3 32 cos(3 ) 0 2π .y x x x x= − = =from to

For Calculator use window [0,2] x [-1,4]

( )( )

( ) ( ) ( ) ( )

3

3

3 3

1/ 3

22 3

1

2 2 3

1

2 22 2 3

1 1

(2 )33

0

31/ 3 33 31/ 3

2

2

2 cos(3 )

2 cos(3 )

2 cos(3 )

2 sin(3 )3 9

2 (2 ) 2 0sin(3 (2 ) ) sin(3 0 )3 9 3 9

4 13

π

.1593

b

a

V x x dx

x x dx

x dx x x dx

x x

V r dx

π

π

π π

π

π

π

π π

π π

π π ππ ππ

π

= −

= −

= −

= −

= − − −

= ≈

=

Find the exact volume of the solid of revolution

formed by rotating the region bounded by the graphs of

about the x-axis on [0, 5]. Sketch.

65 sin(2 ) 2 cos(2 ).y x and y xπ π= + = +

___________V =

Which method is this?______

( )( )

( )( )( )

5 2 2

0

5 2 2

0

5

05

05

0

π ( ) ( )

π 5 sin2 2 cos2

π 5 sin2 2 cos2

π 3 sin2 cos2

cos2 sin23 2 215

V f x g x dx

V x x dx

V x x dx

x x dx

x xx

π π

π π

π π

π ππ π ππ

= −

= + − +

= + − +

= + −

= − − =

Find the exact volume of the solid

of revolution formed by rotating the

region bounded the graphs of

7

2( ) 3 ( )f x x g x x= =and about the y - axis

Sketch.

( )2 2π ( ) ( )d

cV f y g y dy= −

( )2

29

0

29

0

932

0

33 32 2

0

π 3

π 9

π 2 27

(9) (9) (0) (0) 27ππ π2 27 2 27 2

y

y

yV y dy

yV y dy

y yV

V

=

=

= −

= −

= −

= − − − =

What is g(y), the inner radius?________________

What is f(y), the outer raius?________________

Use the washer method.

What is the formula?______

: ( ) 3

: ( )

ySolve forx f y x

Solve forx g y x y

=

=

What are points of intersection?

What are the limits of integration? y = ____ Y = ____

Name_______________

0 9

Washer

Suppose you have to determine the exact volume of

the solid of revolution by revolving the area bounded by

the parabola x = 4 - 3y - y2 and

x = 0 and the x-axis about the x-axis.

It appears that the disk method would be the easiest

method, but when looking at

the diagram and trying to

sub into the formula

we realize that the parabola is not a function and it would

be very difficult to solve for y= f(x). Try the Shell method.

8

r

2π ( )b

aV f x dx =

02π ( ) ( )a

around yV r x h x dx= Since the shell is revolved

around the x- axis we must

2π ( ) ( )d

around x cV r y h y dy= adjust the formula.

r

The radius r(y) the height of the curve, y, and h(y) is thelength of the cylinder. Find the volume.

h(y)

1 2

02π (4 - 3 - )around x y

V y y y dy=

= 1 2 3

0

12 43

0

2π (4 - 3 - )

2π 2 4

3π2

V y y y dy

y yV y

V

=

= − −

=

r(y)

2π ( ) ( )d

cV r y h y dy=

Horizontal Axis of Rotation

( )h y

c

d

( )r y

2π ( ) ( )b

aV r x h x dy=

Vertical Axis of Rotation

b

( )h x

a ( )r x

( )h y

c

d

( )r y

b

( )h x

a ( )r x

Find the exact volume of the solid ofrevolution formed by rotating the region bounded by the graph of x = 9 - y2 and the y- axis on (0 < y < 3) about the x – axis. Sketch 3 2

0

3 3

0

32 4

0

2π (9 - )

2π (9 - )

92π 2 4

81π2

V y y dy

V y y dy

y yV

V

=

=

= −

=

Horizontal

Axis of Rotation

Vertical

Axis of Rotation

©2018 joan kessler distancemath.com

Name_______________