Aim: Shell Method for Finding Volume Course: Calculus Do Now: Aim: How do we find volume using the Shell Method? Find the volume of the solid that results

Aim: Shell Method for Finding Volume Course: Calculus

Do Now:

Aim: How do we find volume using the Shell Method?

Find the volume of the solid that results when the region bounded by x = y3 and x = y2 from y = 0 to y = 1 is revolved about the y-axis.


The Shell Revolution

w

h

p

2w

2w

2w

2w

p

2w

p

2w

p

2w

p 2w

p Axis of Revolution

h


The Shell Method

w

h

2w

p

2w

p

2wp Outer radius

2wp Inner radius

(volume of cylinder)Volume of shell = minus ( )

(volume of hole)


The Shell Method

(volume of cylinder)Volume of shell = minus ( )

(volume of hole)

1

2n

i ii

V p y h y y

2d

cV p y h y dy

take to limit

2 2

2 222 average radius height thickness

w wV p h p h

phw

outer radius

inner radius


To find the volume of a solid of revolution with the shell method

The Shell Method

Horizontal Axis of Revolution

2d

cV p y h y dy

c

d


( )p y

y

h(y)

Vertical Axis of Revolution

2b

aV p x h x dx

Vertical Axis of Revolution

x

h(x)

a b

p(x)


Model ProblemFind the volume of the solid of revolution formed by revolving the region bounded by y = x – x3 and the x-axis (0 < x < 1) about the y-axis.

0.4

0.2

0.5 1

f x = x-x3

p(x) = x

h(x) = x – x3

2b

aV p x h x dx Vertical Axis of Revolution

1 3

02V x x x dx

15 31 4 2

00

2 25 3x x

V x x dx

1 1 42 0 05 3 15

V


Model ProblemFind the volume of the solid of revolution formed by revolving the region bounded by and the y-axis (0 < y < 1) about the x-axis.

2yx e

1

0.5

1

f y = e -y2

h(y)2ye

p(y)


2d

cV p y h y dy

2 1

0

11 1.986

ye

e

21

02 yV ye dy


Comparing Disc and Shell Methods

Disc Method

Shell Method

3

2

1

-1

2

x

Rc

d

r

3

2

1

-1

2

ab

2b

aV phdx

x

p

h

3

2

1

-1

2

y

c

d

p

h

2d

cV phdy

3

2

1

-1

2a bR

ry

2 2d

cV R r dy 2 2b

aV R r dx


Do Now:

Aim: How do we find volume using the Shell Method?

The bases of a solid is the region enclosed by the graphs of y = 1/2x2 and y = 8. Cross sections perpendicular to the y-axis are semicircles with diameter in the plane of the region. Find the volume of the solid.

10

8

6

4

2

-5 5

g x = 8

f x = 0.5x2

diameter


Do-Now

21 2 radius of semi-cirlces2

y x x y

212

A r

21 22

A y

Volume = ( )b

aA y dy

28

0

1 22

y dy

32 cubic units


-1 1

2.5

2

1.5

1

0.5

A: (1, 2)

f x = x2+1

Recall: Model Problem

2

Find the volume of the solid formed by revolving the region bounded by the graphs of 1, 0, 0, and 1 about the -axis.

y x y xx y

y

y

r

R

1 2 2

0

22 2

1

1 0

1 1

V dy

y dy

disk

washer

1 2

0 11 2V dy y dy

22

1

01

22

1 34 2 22 2

yy y

disk

washer

0, 0 1( )

1, 1 2

yr y

y y

1R


-1 1

2.5

2

1.5

1

0.5

A: (1, 2)

f x = x2+1

Shell Method is Preferable

2

Find the volume of the solid formed by revolving the region bounded by the graphs of 1, 0, 0, and 1 about the -axis.

y x y xx y

p(x)

h(x) = x2 + 1

= x

2b


1 2

02 1V x x dx

14 2

0

24 2

3 324 2

x x


2

1

-1

-2

-5 5

f x = 1-x2

16

Model Problem

A pontoon is to be made and is designed by rotating the graph of

about the x-axis, where x and y are measured in feet. Find the volume.

2

1 , 4 416x

y x

8 ft.

2 ft.

Disk method a. rev

Shell method a. rev




2

1

-1

-2

-5 5

f x = 1-x2

16

Model Problem – Disk Method2

1 , 4 416xy x

2 2( ) ( )b

aV R x r x dx

R(x) = 2

116x

r(x) = 0

224

41

16x

V dx

2 44

41

8 256x x

dx

43 5

4

64 cubic feet24 1280 15x x

x




2

1

-1

-2

-5 5

f x = 1-x2

16

Model Problem – Shell Method2

1 , 4 416xy x

p(y) = y

h(y)

4 1x y


2d

cV p y h y dy

1

02 8 1V y y dy

8 1 y

2( ) = h(y)

64 cubic feet15


3

2.5

2

1.5

1

0.5

-1 1 2 3

h(x) = x3 + x + 1

g y = 2

Model Problem – Only Shell

Find the volume of the solid of revolution formed by revolving the region bounded by the graphs of y = x3 + x + 1, y = 1, x = 1 about the line x = 2.

Axis of Revolution

= 2 – x p(x)

(1, 3)

- ( 1)

can’t solve for x


3

2.5

2

1.5

1

0.5

-1 1 2 3

h(x) = x3 + x + 1

g y = 2

Model Problem – Only ShellFind the volume of the solid of revolution formed by revolving the region bounded by the graphs of y = x3 + x + 1, y = 1, x = 1 about the line x = 2.

Axis of Revolution

p(x) = 2 – x

(1, 3)

- ( 1)

2b


1 3

02 2 1 1V x x x dx

1 4 3 2

02 2 2x x x x dx

15 4 32 2

0

1 1 1 292 2 15 2 3 5 2 3 15x x x

x

can’t solve for x


The Shell Method

Documents

Aim: Shell Method for Finding Volume Course: Calculus Do Now: Aim: How do we find volume using the Shell Method? Find the volume of the solid that results