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[Double integral] Area of a triangle

  1. Dec 14, 2014 #1
    Hi! I'm stuck with the following problem:

    -----------------------------------
    Calculate

    ∫∫ (x-y)*|ln(x+2y)| dxdy

    where D is the triangle with corners in the coordinates (0,0), (1,1) and (-3,3)
    -----------------------------------

    I get the following lines that forms the triangle: y=-x, y=x and y=-1/2*x+3/2

    BX4tVDi.jpg

    Im thinking that I start with substitution: (x-y)=u and x+2y=v

    Which gives me ∫∫ u*|ln(v)| dudv

    After that I don't know how to continue. I would like to know which new "intervals" I should do the integral between when I am in my new u- and v-domain?
     
  2. jcsd
  3. Dec 14, 2014 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    That gives ugly borders.
    I would use u=x-y and v=x+y then your triangle stays a nice rectangular triangle.

    In either case, find the coordinates of the corners in the new coordinate system and then set up the integral again.
     
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