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Math Help - Polar Double Integral Problem

  1. #1
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    Polar Double Integral Problem

    Hi, I have been given the task of converting this integral:
    \int_0^1\int_{y^2}^{\sqrt{y}}(x+y) dx dy

    Into polar coordinates. I don't really know where to begin. I found that
    \sqrt{y}=x is equal to tan \theta sec \theta and
    y^2=x is equal to cot \theta csc \theta,
    But I don't know how that might help me. Any help?

    Thanks,
    Peter
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    Re: Polar Double Integral Problem

    Quote Originally Posted by flybynight View Post
    Hi, I have been given the task of converting this integral:
    \int_0^1\int_{y^2}^{\sqrt{y}}(x+y) dx dy

    Into polar coordinates. I don't really know where to begin. I found that
    \sqrt{y}=x is equal to tan \theta sec \theta and
    y^2=x is equal to cot \theta csc \theta,
    But I don't know how that might help me. Any help?

    Thanks,
    Peter
    The first thing you may want to note is that the integrad

    f(x,y)=x+y is symmetric with respect to the line y=x

    The line y=x also symmetrically divides your region of integration. You can use this symmetry to your advantage. You only need to integrate over half of the region upto the line y=x or from the line to the top.

    This will be easier to convert

    2\int_{0}^{1} \int_{y^2}^{x}(x+y)dxdy

    to polar coordinates!

    [img]22704[/img]
    Attached Thumbnails Attached Thumbnails Polar Double Integral Problem-region.jpg  
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