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Math Help - Limits in integral..help!

  1. #1
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    Limits in integral..help!

    The body K is limited by the areas:
    z= x^2 + y^2
    z= 2-x^2-y^2

    Calculate the integral ∫∫∫ (x^2 + y^2) dxdydz

    Now, I should introduce cylindrical coordinates, where upon I get that the body is limited by:

    z= r^2
    z= 2-r^2

    My question is about the limits:

    r goes from ??
    z goes from 2-r^2 to r^2. correct?
    the angle goes from 0 to 2pi

    I would really appreciate some help
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    The projection on the xy plane is x^2+y^2=2-x^2-y^2 or equivalently x^2+y^2=1.

    So, the limits are:

    \begin{Bmatrix}0\leq  \theta\leq 2\pi\\0\leq r \leq 1\\r^2\leq z \leq 2-r^2\end{matrix}

    Regards.

    Fernando Revilla
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