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Homework Help: Triple integral over a sphere in rectangular coordinates

  1. Apr 1, 2008 #1
    1. The problem statement, all variables and given/known data

    Evaluate the following integral:

    [tex]
    \iiint \,x\,y\,z\,dV
    [/tex]

    Where the boundaries are given by a sphere in the first octant with radius 2.

    The question asks for this to be done using rectangular, spherical, and cylindrical coordinates.

    I did this fairly easily in spherical and rectangular coordinates, except for the fact that I got two different answers and I can't figure out where I went wrong! That's not a problem though because I can fix that.


    3. The attempt at a solution

    How would I do this problem in rectangular coordinates? My integral would look like this:

    [tex]
    \int_{0}^{{\sqrt{4-x^2-y^2}}}\int_{0}^{{\sqrt{4-x^2-z^2}}}\int_{0}^{{\sqrt{4-z^2-y^2}}}xyz\,dz\,dy\,dx
    [/tex]

    Which, without some clever transformations and an extremely messy Jacobian calculation, looks unsolvable.
     
  2. jcsd
  3. Apr 1, 2008 #2
    I think you need to rethink your bounds on that one...
     
  4. Apr 1, 2008 #3
    How does this look then?

    [tex]
    \int_{0}^{2}\int_{0}^{2}\int_{0}^{{\sqrt{4-z^2-y^2}}}xyz\,dz\,dy\,dx
    [/tex]
     
  5. Apr 1, 2008 #4
    Hmm, MATLAB tells me that's zero.
     
  6. Apr 1, 2008 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Your dy limit should depend on x.
     
  7. Apr 2, 2008 #6

    HallsofIvy

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    Science Advisor

    That would be over a square in the xy-plane rising up to the sphere.

    Projecting the sphere into the xy-plane gives you the quarter circle x2+ y2= 4, with [itex]0\le x\le 2[/itex], [itex]0\le y\le 2[/itex]. You can let x go from 0 to 2 but then, for each x, y ranges from 0 to [itex]\sqrt{4- x^2}[/itex].
     
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