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Math Help - Double integrals

  1. #1
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    Post Double integrals




    We are learning about double integrals in 15.5. I think i understand the concepts of the chapter, but i really just don't see how to solve this problem. Any help would be greatly appreciated
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  2. #2
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    x^2+ y^2= a^2 is, of course, the circle with center at the origin and radius a. The line x+ y= a goes through (a, 0) and (0, a). This problem is ambiguous as there are two regions bounded by those. I am going to assume they the smaller region, to the upper right of the line.
    For that region, x obviously varies from 0 to a and, for each a, y varies from y= a- x to y=\sqrt{a^2- x^2}. Your integrals should be set up as
    \int_{x=0}^a \int_{y= a- x}^{\sqrt{a^2- x^2}} f(x,y)dydx.

    Do you have to formulas for finding the centroid (in section 15.5)? You should be able to use "symmetry" to cut your work in half.
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  3. #3
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    the only equation is for the moment of inertia. i'm not sure that will help in this example though.
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  4. #4
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    oh actually there is. the center of mass is given by the second eqn here:

    16.5 Applications of Double Integrals.htm
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  5. #5
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    Good! In this problem, since the density is constant, say " \rho", the mass is just
    M= \rho\int_{x= 0}^a\int_{y= a-x}^{\sqrt{a^2- x^2}} dydx

    and then
    M\overline{x}= \rho\int_{x= 0}^a\int_{y= a-x}^{\sqrt{a^2- x^2}} xdydx
    and
    M\overline{y}=  \rho\int_{x= 0}^a\int_{y= a-x}^{\sqrt{a^2- x^2}} ydydx

    The " \rho"s will cancel.
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  6. #6
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    Accidental double post
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