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Volume of the following

  1. Apr 28, 2009 #1
    1. The problem statement, all variables and given/known data

    Let W be the solid bounded by the paraboloid x = y^2 + z^2 and the plane x = 9.

    How can I find the volume of this?

    2. Relevant equations



    3. The attempt at a solution

    I find it hard to visualize so it makes me harder to find the volume... can someone help me?
     
  2. jcsd
  3. Apr 28, 2009 #2

    Hello Equinox. The key to doing these effectively is to become proficient at plotting them. I think in Mathematica you can use:

    ContourPlot3D[x==y^2+z^2,{x,-5,5},{y,-5,5},{z,-5,5}]

    Ok, assume that gets the part you want. Next is to make it transparent so that you can better see the surfaces. Use "PlotStyle->{Opacity[0.5],LightPurple} or something like this.

    Next is to draw the plane x=9. Do that with Polygon command and again use an opacity factor. Then combine the plots with Show[{p1,p2}]. Alright, the learning curve is slow at first, but once you get the hang of it, you can create a very nice visualization relatively quickly of this and much more complicated volume integrations and then the integrations become easy to set up once you have a nice picture. It's worth the effort. :).
     
  4. Apr 28, 2009 #3
    Seems pretty complex.. I don't have mathematica as well
     
  5. Apr 28, 2009 #4

    tiny-tim

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    Hi -EquinoX-! :smile:

    (try using the X2 tag just above the Reply box :wink:)
    It's a paraboloid …

    you can tell because x= r2 is a parabola, so x = y2 + z2 is a parabola rotated about its principal axis. :wink:

    Anyway, just divide the volume into circular slices of thickness dx, find the volume of each slice, and integrate. :smile:
     
  6. Apr 28, 2009 #5
    I am pretty confused... should I integrate in polar, cylindrical, or spherical coordinate?
     
  7. Apr 28, 2009 #6

    tiny-tim

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    Hi -EquinoX-! :smile:

    I think you're making this over-complicated …

    this isn't a ∫∫∫, where you have to decide whether it's dx dy dz or dr dθ dφ or dr dθ dz …

    it's only a single ∫ because you know what the area of a circle is! :wink:

    so, as I said, just use slices of thickness dx, and integrate (over dx) :smile:
     
    Last edited: Apr 28, 2009
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