Change of variable

  1. How do I evaluate the triple integral [tex]\int\int\int_G[/tex] x+y+z dV using a suitable change of variable where G is the region
    0 [tex]\leq[/tex] x+y [tex]\leq[/tex] 1, 2 [tex]\leq[/tex] y+z [tex]\leq[/tex] 3, 4 [tex]\leq[/tex] x+z [tex]\leq[/tex] 5.
    I know to let u = x+y, v = y+z, w = x+z and I end up with the
    det(jac) = |2| [tex]\Rightarrow[/tex] 1/det(jac) = |1/2|. But I'm stuck after that. Help.
     
  2. jcsd
  3. tiny-tim

    tiny-tim 26,055
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    Hi squenshl! :wink:
    Well, you've got the bounds, and you know how to rewrite the dV (from the Jacobian), so all you need is to rewrite x+y+z in terms of u v and w, which is … ? :smile:
     
  4. arildno

    arildno 12,015
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    Gold Member

    Hint:

    What does u+v+w equal, in terms of x+y+z?
     
  5. u+v+w = 2x+2y+2z = 2(x+y+z),
    [tex]\Rightarrow[/tex] x+y+z = (u+v+w)/2.
    Then just chuck that in. Is that right. Thanks.
     
    Last edited: Aug 16, 2009
  6. tiny-tim

    tiny-tim 26,055
    Science Advisor
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    :biggrin: Woohoo! :biggrin:
     
  7. Cheers.
     
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