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    Moment of inertia and Change of variables

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    Using the coordinate change u=xy, v=y/x, set up an iterated integral for the polar moment of inertia of the region bounded by the hyperbola xy=1 , the x-axis, and the two lines x=1 and x=2.
    Choose the order of integration which make the limits simplest

    I found something , I just want you to help on it : here is what I have:

    the region is attached to the x axis...
    so the polar momentum is iqual to:
    double integral(y^2)dxdy
    ---->but y^2=u*v
    AND the jacobian matrice of U... and v...
    is equal to 2*y/x=2*v
    so the inverse is 1/(2v)

    THEN WE CAN SAY that
    "double integral(y^2)dxdy=(1/(2v))*uv.dvdu" right?

    when x=1 and y=0 we get u=v
    when xy=1 we get u=1
    finally when x=2 then v=u/4 right?

    so we see that we have

    double integral(u/2)dudv, for u varying from 1 to v
    and v varying from 1 to v, no?

    My result then is "1/24(1-v^3)"

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    Solution Summary

    The solution demonstrates how to apply change of variables to a double integral, from the change in the integrand to the change in integration limits.
    The solution contains 3 pages of completre derivations.