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Math Help - Triple integral..confused about limits

  1. #1
    Junior Member
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    Sep 2010
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    Triple integral..confused about limits

    Hello!

    I need help with the following task:
    ∫∫∫ z/(1+x^2+y^2) dxdydz
    D

    where D is: {(x,y,z); x^2+ y^2+z^2 equal to or less than 1, z equal to or bigger than sqrt(x^2 + y^2)}

    I think I need to introduce polar coordinates, but I don't understand which limits to set..can someone please help me with this? I am preparing for an exam next week and really need some guidance.

    Cheers!
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  2. #2
    MHF Contributor

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    x^2+ y^2+ z^2\le 1 is inside and on the unit sphere. \sqrt{x^2+ y^2}= z is the upper nappe of a cone so the region here is above that cone but inside the sphere. Yes, convert to polar coordinates (strictly speaking cylindrical coordinates). x= r cos(\theta), y= r sin(\theta), so the sphere is r^2+ z^2= 1 and the cone is r= z. The cone and the sphere intersect where r^2+ z^2= r^2+ r^2= 2r^2= 1 or r= \sqrt{\frac{1}{2}}= \frac{\sqrt{2}}{2}. The integral with respect to \theta is from 0 to 2\pi, of course. The integral with respect to r is from 0 to \frac{\sqrt{2}}{2} and the integral with respect to z from r to \sqrt{1- r^2}.
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