Section 7.3 – Volume: Shell Method

White Board ChallengeCalculate the volume of the solid obtained by rotating the region bounded by y = x2, x=0, and y=4 about x = -1.

Calcu

lator

24 2

0

561 0 1

3y

We will now investigate another method to calculate this volume.

Volume of a ShellConsider the following cylindrical shell (formerly a washer):

routerh

r inn

er 2

outer innerr rR

2C R

The average of the radii is a new radius from the center of

the base to the middle of the

enclosed area.

R

Imagine the circle in in the middle of the

base area.

Label the new radius.

Thus, the circumference of the

middle circle is…

outer innerr r r Also, the thickness of

the shell is…

Δr

Volume of a ShellThe volume of the cylindrical shell is easier to see when it is flattened out:

2V R h r

h

C = 2πR

The cylindrical shell flattened out is a

rectangular prism.The length of the

base is…

The height of the base is…

The height of the prism

is…

ΔrThus the volume of the prism is…

Volumes of Solids of Revolution with Riemann Sums

Let us rotate the region under y=f(x) from x=a to x=b about the y-axis. The resulting solid can be divided into thin concentric shells.

Volume

2 R h t 1

n

kmax 0

limkx

2b

aR h dx

a b

Height

2b

aR h dx

thickness

radius

Volumes of Solids of Revolution: Shell Method

• Sketch the bounded region and the line of revolution. • If the line of revolution is horizontal, make sure the

equations can easily be written in the x= form. If vertical, the equations must be in y= form.

• Sketch a generic shell (a typical cross section).• Find the radius of the generic shell (perpendicular

distance from the line revolution to the outer edge of the shell), the height of the shell (this length is perpendicular to the radius), and the thickness of the shell (this length is perpendicular to the height).

• Integrate with the following formula:

2b

aV radius height thickness

Opposite of Washer Method

MA

KE

A H

OO

K:

Example 1Calculate the volume of the solid obtained by rotating the region bounded by y = x2, x=0, and y=4 about x = -1.

Sketch a Graph

Find the Boundaries/Intersections

Make Generic Shell(s)

Height = 4 – x2

Integrate the Volume of the Shell

2 2

02 1 4x x dx

56

3

2x

2 4x 0x x

We only need x>2

Line of Rotation

Radius = x – -1 = x + 1

Thickness = dx

Example 2Calculate the volume V of the solid obtained by rotating the region bounded by y = 5x – x2 and y = 8 – 5x + x2 about the line y-axis.

Sketch a GraphFind the Boundaries/Intersections

2 25 8 5x x x x

Make Generic Shell(s)

Height =

( 5x – x2 ) – (8 – 5x + x2)

Integrate the Volume of Each Generic Washer

4 2 2

12 5 8 5x x x x x dx

45

1, 4x

Lin

e o

f R

ota

tio

n

Radius = x

Thickness = dx

White Board ChallengeUse the shell method to calculate the volume of the solid obtained by rotating the region bounded by y = x1/2 and y=0 over [0,4] about the x-axis.

Calcu

lator

2 2

02 4y y dy

Lin

e o

f R

ota

tio

n

Radius = y

Height = 4 – y2

Thickness = dy

8