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Homework Help: Calculate ∫∫x^2 dS

  1. Jan 3, 2012 #1
    1. The problem statement, all variables and given/known data

    Calculate ∫∫x^2 dS where S is the triangle with corners (1,1,0) (0,1,0) and (0,0,1)

    The triangle is the graph of g(x,y)=1-x-y

    One integral can be [itex]\displaystyle \sqrt3 [\int_{0}^{1}(\int_{0}^{1-x}x^2dy)dx][/itex]

    I calculate the other one to be

    [itex]\displaystyle \sqrt3 [\int_{0}^{1}(\int_{0}^{1-y}(1-y)^2dx)dy][/itex]

    but I dont get the same answer...can some one point it out?

  2. jcsd
  3. Jan 3, 2012 #2


    Staff: Mentor

    Re: Integrals

    The integrand, x2, should not change when you change the order of integration.
  4. Jan 3, 2012 #3
    Re: Integrals

    OK, thanks. I am wondering did I come across situations where one does change the integrand or perhaps I am confusing it with changing the limits when there is a u substitution involved etc?

  5. Jan 3, 2012 #4


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Re: Integrals

    I'm pretty sure the triangle is in the plane, y+z = 1 .
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