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Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the curves 𝑦=𝑥3,y=x3, 𝑦=8,y=8, and 𝑥=0x=0 about the 𝑥x-axis.

 Feb 21, 2020
 #1
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See the graph  for this region : https://www.desmos.com/calculator/q9jn22a8q4

 

Because we are rotating about the x axis,  the area will be a function of  y.....so.... we  need  to  write y  =x^3   as    x  = y^1/3

 

So  we  have

 

      8

2pi ∫  radius  *  height   dy      =

      0

 

       8

2pi  ∫  y  * y^(1/3)  dy   =       

       0

 

      8

2pi ∫   y ^ (4/3)   dy    =

     0

 

                                 8

2pi   * [ ( 3/7) y^(7/3) ]        =

                                 0

 

(6 pi / 7 )  *  [  ( 8^(7/3)   -  0^(7/3) ]  =

 

(6 pi /7) * [  8^(1/3) }^7   =

 

(6pi /7)  *  [ 2^7 ]  =

 

(6 pi / 7) * 128     ≈     344.68  units^3

 

 

 

cool cool cool

 Feb 21, 2020

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