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Find the volume of the region bounded by parabolic cylinder and planes

  1. Apr 9, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the volume of the solid bounded by the parabolic cylinder y = x^2 and the planes z = 3-y and z = 0


    2. Relevant equations



    3. The attempt at a solution
    Obviously, a triple integral must be used in the situation. Our professor never explained how to find the limits of integration, this is the part that is confusing me. I believe that the limits for z are 3-y and 0. However, I have no idea how to get the limits of x and y. Any help is greatly appreciated.
     
    Last edited: Apr 9, 2012
  2. jcsd
  3. Apr 9, 2012 #2

    tiny-tim

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    hi mharten1! :smile:
    no, your first limits (for z, say) will just be two numbers (no y)

    your next limits (for y, say) will depend on z

    your third limits will depend on y and z

    standard method: cut the region into horizontal slices between heights z and z+dz …

    what is the 2D shape of each slice? :wink:
     
  4. Apr 9, 2012 #3

    Dick

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    Basically you want to draw a sketch of the solid you are trying to find the volume of. Can you visualize it? z=0 is the x-y plane. Where does the plane z=y-3 intersect the x-y plane?
     
  5. Apr 9, 2012 #4
    Doesn't it intersect it when z=0? So at y=3?
     
  6. Apr 9, 2012 #5

    Dick

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    Sure. So what region do you want to integrate z over in the x-y plane?
     
  7. Apr 9, 2012 #6
    The region from 0 to 3? If that's not right, I guess I'm not visualizing this in the right way. I think I'll graph out the region so I can better see it.
     
  8. Apr 9, 2012 #7

    Dick

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    I think you should graph it out. You should anyway. "The region from 0 to 3?" isn't a very accurate description.
     
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