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Applications of Double Integrals: Centroids and Symmetry

  1. Jun 7, 2012 #1
    1. The problem statement, all variables and given/known data
    A lamina occupies the region inside the circle x2+y2=2y but outside the circle x2+y2=1. Find the center of mass if the density at any point is inversely proportional to its distance from the origin.

    Here is the solution:
    https://dl.dropbox.com/u/64325990/MATH%20253/Centroids.PNG [Broken]

    Why does it say by symmetry of the region of integration. Shouldn't it be by symmetry of the density function p(x,y) = k/root(x2+y2)?

    For example what if our p(x,y) = x. Even though the region D is symmetric, the mass is no longer symmetric and the balancing point is no longer at x = 0. Am I right?

    Thanks
     
    Last edited by a moderator: May 6, 2017
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  3. Jun 7, 2012 #2

    SammyS

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    Yes, you are correct !
     
    Last edited by a moderator: May 6, 2017
  4. Jun 7, 2012 #3
    To reinstill faith in your textbook, it could be read as:

    Symmetry of D,

    and since f(x)=x.

    Or even that f(x)=x is antisymmetric, perhaps a type of symmetry.
     
  5. Jun 7, 2012 #4

    vela

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    You need the symmetry of both ##\rho(x,y)## and D. The density ##\rho(x,y)## is generally positive, so the only symmetry you can have is even symmetry. When multiplied by x, you get an odd integrand which then integrates to 0 because D is symmetric.
     
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