# Volumes of Revolution The Shell Method by slJhjcH0

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```									Volumes of Revolution
The Shell Method
Lesson 7.3
Shell Method
• Based on finding volume of cylindrical
shells
   Add these volumes to get the total volume
• Dimensions of the shell
 Thickness of the shell
 Height

2
The Shell
• Consider the shell as one of many of a
solid of revolution      dx

f(x)
f(x) – g(x)           x

g(x)

• The volume of the solid made of the sum
of the shells        b
V  2  x  f ( x)  g ( x) dx
a
3
Try It Out!
• Consider the region bounded by x = 0,
y = 0, and y  8  x2

2 2
V  2        x  8  x dx
2

0                     4
Hints for Shell Method
• Sketch the graph over the limits of integration
• Draw a typical shell parallel to the axis of
revolution
• Determine radius, height, thickness of shell
• Volume of typical shell

2    radius  height  thickness
• Use integration formula
b
Volume  2    radius  height  thickness      5
a
• Rotate the region bounded by y = 4x and
y = x2 about the x-axis
thickness = dy

• What are the dimensions needed?                             y
height =   y
    y
16
   height              V  2  y   y   dy
   thickness                  0         4
6
• Possible to rotate a region around any line
f(x)             g(x)

x=a

• Rely on the basic concept behind the shell
method
Vs  2    radius  height  thickness
7
• What is the radius?                               r
g(x)
f(x)
a–x

• What is the height?                         x=c
x=a
f(x) – g(x)

• What are the limits?
c<x<a

• The integral:             a
V   (a  x)  f ( x)  g ( x) dx
c                                       8
Try It Out
• Rotate the region bounded by 4 – x2 ,
x = 0 and, y = 0 about the line x = 2

r=2-x
4 – x2

2

• Determine radius, height, limits      
0
9
Try It Out
• Integral for the volume is
2
V  2  (2  x)  (4  x 2 ) dx
0

10
Assignment
• Lesson 7.3
• Page 277
• Exercises 1 – 21 odd

11

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