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Iterated Integrals bounded by curves

  1. Sep 17, 2007 #1
    Evaluate [tex]\int[/tex][tex]\int_{Q}\left(1 - x^{3}\right)y^{2} dA[/tex] where Q is the region bounded by y=x^2 and x = y^2

    So I have drew the graphs of y=x^2 and x=y^2 and found that they intersect at (0,0) and (1,1). Now I am confused what to replace Q with, but I think it should be this: please tell me if I am incorrect in my selection.

    [tex]\int^{1}_{0}[/tex][tex]\int_{\sqrt{y}}^{y^{2}}\left(1 - x^{3}\right)y^{2} dx dy[/tex]

    or should I be integrating w.r.t y first? also have I mixed up the y^2 and the sqrt(y) in the limit of integration?
  2. jcsd
  3. Sep 17, 2007 #2


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    dxdy is correct, but sqrt(y) > y^2.
  4. Sep 17, 2007 #3
    how would i know this for future reference, i am having serious trouble with the limit of integration part.
  5. Sep 17, 2007 #4


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    Well, it's clear that either [itex]\sqrt{y} < y^2[/itex] for every y in that interval, or [itex]y^2 < \sqrt{y}[/itex] for every y in that interval, correct?

    So, if you try one actual value of y in that interval...

    A little algebra would solved it too: what happens if you manipulate that inequality to put all of the y's on the same side?
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