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
   Radius 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
          Rotation About x-Axis
• Rotate the region bounded by y = 4x and
  y = x2 about the x-axis
                  thickness = dy



                          radius = y


• What are the dimensions needed?                             y
                                                height =   y
     radius                                                  4
                                           y
                                   16
     height              V  2  y   y   dy
     thickness                  0         4
                                                              6
Rotation About Noncoordinate Axis
• 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
Rotation About Noncoordinate Axis
• 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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