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Homework Help: Volume between two paraboloids on x and y axes.

  1. Mar 29, 2012 #1
    1. The problem statement, all variables and given/known data

    Grr it got all got erased...This is a math problem that a student I tutor got wrong on his calculus test. I am having problems remembering what to do. I have looked up other problems, but the paraboloids seem simpler for some reason. Probably because someone is explaining it to me.

    Find the volume between the two parabaloids x = [itex]y^{2}[/itex] + [itex]z^{2}[/itex] and y = [itex]x^{2}[/itex] + [itex]z^{2}[/itex]

    Hint: x - y divides [itex]x^{2}[/itex] - [itex]y^{2}[/itex] (I do not know how this helps. Probably because I am doing it wrong)

    3. The attempt at a solution

    Set z = 0 to find the "shadow" in the xy plane: y = [itex]x^{0.5}[/itex] and y = [itex]x^{2}[/itex]

    They intersect at (0,0) and (1,1) because:
    [itex]x^{2}[/itex] = [itex]x^{0.5}[/itex]
    [itex]x^{2}[/itex] - [itex]x^{0.5}[/itex] = 0
    [itex]x^{0.5}[/itex]([itex]x^{1.5}[/itex] - 1) = 0
    [itex]x^{0.5}[/itex] = 0 gives x = 0 and y = 0 and
    [itex]x^{1.5}[/itex] - 1 = 0
    [itex]x^{1.5}[/itex] = 1 gives x = 1 and y =1

    I think I understood up until now, but am sort of lost, and I do not have the answer given by the teacher to check myself. Do I do ∫∫∫ 1 dzdydx where 0≤x≤1, [itex]x^{2}[/itex]≤y≤[itex]x^{0.5}[/itex] and for the interval of z I just solve one of the given paraboloids for z, which will give a ±square root? Like z = ±√(x - [itex]y^{2}[/itex]) Then I would do trig substitution? Or would I subtract right from left so that I would just do a double integral of √(x - [itex]y^{2}[/itex]) - √(y - [itex]x^{2}[/itex]) with the above intervals for x and y?
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
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