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Math Help - Find the volume of a region using an iterated integral

  1. #1
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    Find the volume of a region using an iterated integral

    The region Q is bounded by z=2x^2+y^2 and z=2-x^2-y^2

    Ok, I tried using Cartesian coords and got a big mess. Then I tried changing everything to polar coords and got this:

    2r^2cos^2(\theta)+r^2sin^2(\theta)=2-r^2

    I was hoping that the left side would turn out as nicely as the right side did. Did I do something wrong when converting? Is there something I'm missing? Thanks
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  2. #2
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    Well, you could write 2r^2cos^2(\theta)+ r^2 sin^2(\theta)= r^2 cos^2(\theta)+ r^2(cos^2(\theta)+ sin^2(\theta)) = r^2(cos^2(\theta)+ 1) and then the equation becomes
    r^2(cos^2(\theta)+ 1)= 2- r^2

    r^2(cos^2(\theta)+ 2)= 2

    r= \frac{\sqrt{2}}{\sqrt{cos^2(\theta)+ 2}}
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