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Math Help - Double integral over strange region

  1. #1
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    Double integral over strange region

    The parabolas x = -y^2, x = 2y - y^2, x = 2 - y^2 - 2y divide the plane into 7 regions, of which only one is bounded. Let this region be A. Find a change of variables such that the first two parabolas become u=0 and v=0. Evaluate the double integral \int_{A} x dx dy.
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  2. #2
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    I should clarify - I can find the Jacobian factor and the "new" integrand in terms of u and v. The only problem is that I don't know what the region should be (ie the limits of the u and v integrals). In fact I'm not too sure how to desribe the region on x and y coordinates (ie I'm not sure what the limit of the integrals would have been before the change of variables anyway).
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  3. #3
    Member Mentia's Avatar
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    A way to see it better would to graph the 3 curves. Check out the region 0<y<1 , -2<x<2.

    So the integration in that region would be, in x and y coordinates:

     \int_{0 }^{ \frac{1 }{2 }  }  \int_{ -y^{2}}^{2y-y^{2} }xdxdy +  \int_{ \frac{ 1}{ 2}  }^{ 1  }  \int_{ -y^{2}}^{2-y^{2}-2y }xdxdy

    Darnit! For some reason it wont let me edit my own posts. My integrals are slightly wrong, they should be dxdy instead of dydx

    [Fixed. -K.]
    Last edited by Krizalid; March 7th 2009 at 02:28 PM.
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