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Variable substitution problem

  1. Nov 10, 2008 #1
    1. The problem statement, all variables and given/known data
    [tex]\int\int\int _E\(x^2y}\;dV[/tex]

    Where E is the solid bounded by [tex]x^2/a^2+y^2/b^2+z^2/c^2=1[/tex]


    2. Relevant equations

    variable substitution x=au, y=bv, z=cw.

    3. The attempt at a solution

    I found the jacobian (abc) and I substituted my variables but I can't find the limits of integration. The only equation I have for the limits is [tex]u^2+v^2+w^2\leq1[/tex]. I don't know how to find the limits of integration for u, v, and w individually.
     
  2. jcsd
  3. Nov 10, 2008 #2

    Dick

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    Now switch to spherical coordinates in u,v,w.
     
  4. Nov 10, 2008 #3

    HallsofIvy

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    First decide on the order in which you want do integrate:
    [tex]\int dudvdw[/tex]?

    Fine. Project the figure on to the vw plane: v2+ w2= 1. Then project that onto the w line: the segment from w= -1 to 1. The limits on the outer "dw" integral have to be numbers. In order to cover the entire figure, w must vary from -1 to 1. For each w, then v must vary from [itex]-\sqrt{1- w^2}[/itex] to [itex]\sqrt{1- w^2}[/itex]. Finally, for each v and w, u varies from [itex]-\sqrt{1- v^2- w^2}[/itex] to [itex]\sqrt{1- v^2- w^2}[/itex]. Those are the limits of integration.

    Of course, because u2+ v2+ w2= 1 is a sphere in uvw-space, spherical coordinates, as Dick suggested, are simplest. The limits of integration would be exactly the same as if it were x2+ y2+ z2= 1.
     
  5. Nov 10, 2008 #4
    oh, ok thanks i didn't even think about switching to polor coordinates.
     
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