6.3 1

Section 6.3: Volumes By Cylindrical Shells

• Objective– Understand how to find the volume of a solid of

revolution using the method of cylindrical shells

6.3 2

Problem

2 32y x x= −

Find the volume of the solid obtained by rotating about the y axis the region bounded by

2 32 0y x x and y= − =

6.3 3

Discussion

2 32y x x= −

6.3 4

Discussion

2 32y x x= −

If we tried to use the washer method, we would have to solve for x in terms of y. This is not possible with the tools we have. There is a way out.

6.3 5

The Shell Methody

a b

y=f(x)

R

x

6.3 6

The Shell Method

y

a b

y=f(x)

R

x

Given a continuous nonnegative function f defined on [a,b]. Consider the region R. Revolve R about the y axis. A solid called a cylindrical shell is generated. What is its volume?

6.3 7

Shell Method: General Formula

2 ( ) ( )b

a

V p x h x dxπ= ∫ 2 ( ) ( )d

c

V p y h y dyπ= ∫

p = distance from the center of the

rectangle to the axis of revolution

h =height of the rectangle

6.3 8

Volume of solid: Shell Method Formula

2 ( )b

a

V x f x dxπ= ∫ Rectangular slice in region rotated is parallel to the y axis(axis of rotation)

2 ( )d

c

V y g y dyπ= ∫Rectangular slice in region rotated is parallel to the x axis(axis of rotation)

Typical shell has radius x, circumference 2 ( )x and height f xπ

6.3 9

Steps to find volume using the Shell Method

(a) Sketch the region R

(b) Show a typical rectangular slice properly labeled

(c) Write a formula for the approximate volume of the

shell generated by this slice

(d) Set up the corresponding integral

(e) Evaluate the integral

6.3 10

Example

Find the volume of the solid of revolution formed by revolving the region bounded by the graph of and the y axis

about the x-axis.

2yx e−= 0 1y≤ ≤

6.3 11

6.3 12

Continued 2

( ) yh y e−=

6.3 13

6.3 14

Example

The region bounded by the parabola and the

line y=2 is rotated about the y axis. Find the volume of the

resulting solid.

23y x x= −

6.3 15

Solution

6.3 16

Typed Solution

The region bounded by the parabola and the

line y=2 is rotated about the y axis. Find the volume of the

resulting solid.

23y x x= −

The length of a rectangular slice is ( )23 2h x x= − −

and the distance from middle of the rectangle to the y axis is x

6.3 17

Solution

( )2

2

1

2 3 2V x x x dxπ= − −∫ ( )2

2 3

1

2 3 22

x x x dx ππ= − − =∫

6.3 18

Example: Shell Method; Rotation about line parallel to x –axis.

The region bounded by the line y=1-x, the x-axis, and the y-axis is revolved about the line y=-1.

Find the volume of the solid generated by the (a) shell method (b) washer method

6.3 19

6.3 20

6.3 21

Typed: Solution: Shell Method; Rotation about line parallel to x –axis.

( ) ( )1

0

2 1 1V y y dyπ= + −∫ ( )1

2

0

2 1 y dyπ= −∫13

0

423 3yyπ π

⎛ ⎞= − =⎜ ⎟

⎝ ⎠

6.3 22

Solution: Washer Method; Rotation about line parallel to x –axis.

( ) ( )1

2 2

0

( ) ( )V f x g x dxπ ⎡ ⎤= −⎣ ⎦∫

( ) ( )1

2 2

0

1 ( 1) 0 ( 1)x dxπ= − − − − − −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦∫

( )1

2

0

2 1x dxπ ⎡ ⎤− −⎣ ⎦∫43π

=