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Changing the Order of Integration in this Triple Integral

  1. Dec 10, 2013 #1
    1. The problem statement, all variables and given/known data

    Evaluate ##\iiint_D (x^2+y^2)\mathrm{d}V##, where ##D## is the region bounded by the graphs of ##y=x^2##, ##z=4-y##, and ##z=0##.

    2. Relevant equations



    3. The attempt at a solution

    So after over at least an hour of thinking, I might have all 6 orders of integration. Can someone check my answer please?

    1:
    $$x=-\sqrt{y}\quad to\quad x=\sqrt{y}\\y=0\quad to\quad y=4\\z=0\quad to\quad z=4-y$$

    2:
    $$x=-\sqrt{y}\quad to\quad x=\sqrt{y}\\y=0\quad to\quad y=4-z\\z=0\quad to\quad z=4$$

    3:
    $$y=0\quad to\quad y=4-z\\x=-\sqrt{-z+4}\quad to\quad x=\sqrt{-z+4}\\z=0\quad to\quad z=4$$

    4:
    $$y=0\quad to\quad y=4-z\\x=-2\quad to\quad x=2\\z=0\quad to\quad z=-x^2+4$$

    5:
    $$z=0\quad to\quad z=4-y\\x=-2\quad to\quad x=2\\y=x^2\quad to\quad y=4$$

    6:
    $$z=0\quad to\quad z=4-y\\x=-\sqrt{y}\quad to\quad x=\sqrt{y}\\y=0\quad to\quad y=4$$

    I've been informed 3 and 4 are incorrect. Does anyone happen to know the correct bounds so I can know where I went wrong?
     
  2. jcsd
  3. Dec 10, 2013 #2
    Got two suggestions:

    (1) Your results are poorly-formatted. What's being integrated with respect to what? Not easy to see and when you're asking for help you want to be as clear, concise, and to the point as possible as a courtesy to those helping you. Don't want to format it nicely huh?

    (2) Do you have access to Mathematica or other where you can just check them yourself?
     
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