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Triple Integral - Change the order of integration

  1. Nov 30, 2011 #1
    1. The problem statement, all variables and given/known data

    [itex]\int^{1}_{0}\int^{x^2}_{0}\int^{y}_{0} f(x,y,z) dz dy dx[/itex]

    Find 5 equivalent iterated integrals.

    2. Relevant equations

    [itex]0 ≤ z ≤ y[/itex]

    [itex]0 ≤ y ≤ x^2[/itex]

    [itex]0 ≤ x ≤ 1[/itex]

    3. The attempt at a solution

    1) [itex]\int^{1}_{0}\int^{√y}_{0}\int^{x^2}_{0} f(x,y,z) dz dx dy[/itex]

    I will try dz dy dx first.

    Because y = x^2, so [itex]0 ≤ z ≤ x^2[/itex]

    Because y = x^2, so [itex]0 ≤ x ≤ √y[/itex]

    And by the same logic, [itex]0 ≤ y ≤ 1[/itex]

    When I integrate for f(x,y,z) = 1, the correct answer is 1/10. I do not get the same answer with my solution. Help! Is it possible to solve this without graphing it? Or is it necessary to get the correct answer?
  2. jcsd
  3. Nov 30, 2011 #2
  4. Nov 30, 2011 #3


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    Staff Emeritus
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    Here is an error. For each x, y goes from 0 to [itex]x^2[/itex]. If you graph that region in an xy-plane, it is below and to the right of the graph of [itex]y= x^2[/itex]. That means that, for each y, x goes from [itex]\sqrt{y}[/itex] up to 1. The y-integral is [itex]\int_{x^2}^1 dy[/itex]

    Looking at this, I realize now that I made that mistake in the previous post. I have edited it.

    Last edited: Nov 30, 2011
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