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Find center of mass and coordinates using double integrals?

  1. Feb 18, 2013 #1
    1. The problem statement, all variables and given/known data
    Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ.
    D is bounded by the parabolas y = x^2 and x = y^2; ρ(x, y) = 23√x

    2. Relevant equations

    m = [itex]\int[/itex][itex]\int[/itex][itex]_{D}[/itex] ρ(x, y) dA
    x-bar = [itex]\int[/itex][itex]\int[/itex][itex]_{D}[/itex] x*ρ(x, y) dA
    y-bar = [itex]\int[/itex][itex]\int[/itex][itex]_{D}[/itex] y*ρ(x, y) dA

    3. The attempt at a solution
    I integrated 23√x using order dydx with limits: x^2 ≤ y ≤ √x and 0 ≤ x ≤ 1.
    m = 69/14.

    The problem I'm having is with the coordinates. I first got (x-bar,y-bar) as (14/62, 28/1265) but that was wrong in my online assignment. I used the same limits of integration and the above equations to find x-bar and y-bar. I don't know where I'm going wrong with the coordinates... Am I suppose to be using different limits?
    Last edited: Feb 18, 2013
  2. jcsd
  3. Feb 18, 2013 #2


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    If you are trying to compute m you just integrate 23. Why are you integrating sqrt(x) as well? m isn't 69/14. Can you spell out what you are doing in a little more detail?
  4. Feb 18, 2013 #3


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    Your equations for x-bar and y-bar are incorrect. What you have are the expressions for the moments of D about the x and y axes. In order to find x-bar and y-bar, you must divide the moment values by the mass of the region D.
  5. Feb 18, 2013 #4
    Oh my mistake, I used:

    y-bar = 1/m [itex]\int[/itex][itex]\int[/itex][itex]_{D}[/itex] y*ρ(x, y) dA
    x-bar = 1/m [itex]\int[/itex][itex]\int[/itex][itex]_{D}[/itex] x*ρ(x, y) dA

    to find the coordinates.
  6. Feb 18, 2013 #5


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    And in your case, [itex]\rho(x,y)= 23[/itex], a constant. So
    [tex]\overline{x}= \frac{23}{m}\int\int x dxdy[/tex]
    [tex]\overline{y}= \frac{23}{m}\int\int y dxdy[/tex]
    [tex]m= 23\int\int dxdy[/tex]

    You are given "D is bounded by the parabolas y = x^2 and x = y^2". Those, of course, intersect at x= 0 and at x= 1. What will the limits of integration be?
  7. Feb 18, 2013 #6
    Sorry for the confusion, the question was asking to integrate 23√x. I edited my first post.
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