Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The...

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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

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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= − =

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Discussion

2 32y x x= −

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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.

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The Shell Methody

a b

y=f(x)

R

x

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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?

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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

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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

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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

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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≤ ≤

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Continued 2

( ) yh y e−=

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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= −

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Solution

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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

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Solution

( )2

2

1

2 3 2V x x x dxπ= − −∫ ( )2

2 3

1

2 3 22

x x x dx ππ= − − =∫

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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

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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

⎛ ⎞= − =⎜ ⎟

⎝ ⎠

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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

=