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Math Help - Evaluate the Integral over the given region. BOUNDARIES

  1. #1
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    Evaluate the Integral over the given region. BOUNDARIES

    Here is a picture of the problem:
    Evaluate the Integral over the given region. BOUNDARIES-photo-2.jpg


    I know that the y values are bounded between y=-1 and y=1+x^2.
    I am slightly confused about how the x values are bounded. They are bounded on the left by x=-1 and on the right by x=y^2, but how do I also show that it is bounded by x=1?
    Do I need to evaluate the area by using three integrals?

    Thank you for the help!
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  2. #2
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    I would use three integrals
    \int_{-1}^{1 + x^2}\!\!\!\int_{-1}^0\! xy\,dx\,dy + \int_{\sqrt{x}}^{1+x^2}\!\!\!\int_{0}^1\! xy\,d\,xdy + \int_{-1}^{-\sqrt{x}}\!\!\!\int_{0}^1\! xy\,dx\,dy
    Last edited by lvleph; August 2nd 2010 at 06:27 AM.
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  3. #3
    MHF Contributor

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    Here's another way to do it: The integral over the entire square is [tex]\int_{x= -1}^1\int_{y= -1}^2 xydy[tex].

    Subtract from that the integral above the parabola y= x^2+ 1 and below y= 2, \int_{x= -1}^1\int_{y= x^2+ 1}^2 xy dydx, and the integral from x= y^2 to x= 1, \int_{y= =1}^1\int_{x= y^2}^1 xydxdy.
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  4. #4
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    Perfect! Thanks to you both, its so simple I'm disappointed I never thought of doing it like that!
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