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Math Help - Volume under paraboloid and over a disk

  1. #1
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    Volume under paraboloid and over a disk

    Ok its a simple question really... say that I have to find the volume (using polar coordinates) of the solid under the paraboloid z=x^2+y^2 and above the disk x^2+y^2≤9. My approach would be to find the z value of where the cylinder and paraboloid intersect. Then find the volume of the paraboloid using the value of the plane where it intersects, then subtract it from the volume of the cylinder, this seems right to me. I'm getting z=18 for the point where the cylinder and paraboloid intercept, is this right? For some reason my gut is telling me that its 9 not 18. I simply plugged in the radius of the circle into the paraboloid formula. is there another way that I am missing?
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  2. #2
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    Ok after calculating this I got 40.5*pi, is it correct? Is there a way that I can check this?
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  3. #3
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    Do you mean x^2+y^2=9 or do you mean less than or equal to?
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  4. #4
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    Its correct:

    under the paraboloid z=x^2+y^2

    above the disk x^2+y^2≤9
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  5. #5
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    I got the same answer as you!

    Using polar transformations:

    x=rcos(\theta) and y=rsin(\theta)

    so we have z= (rcos(\theta))^2+(rsin(\theta))^2
    z=r^2

    This turns the integral

    \int x^2 + y^2 dxdy to \int r^2 * rdrd\theta

    and using that we have 0<r<3 from the disc constraint, and we have 0<\theta<2\pi because we want the full circle
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  6. #6
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    Ok thanks, I just wanted to make sure whether I had the correct answer. I wonder if I can check this using the washer method, just to be sure.
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  7. #7
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    I did it using the washer method...and you do get the same answer! I also like checking my answers using other methods!
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