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Thread: Help with definite triple integral.

  1. #1
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    Help with definite triple integral.

    I have to evaluate the following integral;

    $\displaystyle \int_{-2}^{-1}\int_{-1}^{-2}\int_{-2}^{-1} x^2/(x^2+y^2+z^2) dx dy dz$

    I sort of solved the first part (dx) with the integration formula:
    $\displaystyle \int 1/(x^2+a^2) dx = 1/a *tan^{-1}(x/a)$

    I did the following:
    First I established $\displaystyle (y^2+z^2)=a^2$
    $\displaystyle \int_{-2}^{-1} x^2/(x^2+a^2) dx= \int_{-2}^{-1} {1-a^2/(x^2+a^2)} dx= \int_{-2}^{-1} 1 dx -\int_{-2}^{-1} a^2/(x^2+a^2) dx=(-1-(-2)) -a^2*\int_{-2}^{-1} 1/(x^2+a^2) dx$
    $\displaystyle =1 -a^2*(1/a *tan^{-1}(x/a))_{-2}^{-1}=1 -a^2*(1/a *tan^{-1}({-2}/a)-1/a *tan^{-1}({-1}/a))=1 -a*tan^{-1}({-2}/a)+ a*tan^{-1}({-1}/a))$
    Then substitute a back to it's original values:
    $\displaystyle 1 -\sqrt{(y^2+z^2)}*tan^{-1}({-2}/\sqrt{(y^2+z^2)})+ \sqrt{(y^2+z^2)}*tan^{-1}({-1}/\sqrt{(y^2+z^2)}))$

    After this I have no idea what to do.

    Any help would be appreciated but if possible I would like to know how to solve the entire problem.
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  2. #2
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    Re: Help with definite triple integral.

    Quote Originally Posted by jollybenito View Post
    I have to evaluate the following integral;

    $\displaystyle \int_{-2}^{-1}\int_{-1}^{-2}\int_{-2}^{-1} x^2/(x^2+y^2+z^2) dx dy dz$
    Where did this integral come from? Is it an attempt to solve some problem? If so, could you state the original problem for us?
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  3. #3
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    Re: Help with definite triple integral.

    Help with definite triple integral.-captura1.jpg
    It isn't from a real life problem, I have been interested in taking a post graduate course in Science Data, it's from the page of sample questions and it's the only one I haven't been able to solve, I wanted to ask specifically cause I have always had difficulty with integrals that involve trigonometric functions and/or division.
    I annex a picture from the original question, it only states "Calculate the following integral".
    Don't really know how to deal with it, any hint, help or reference to a similar problem would be highly appreciated.
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  4. #4
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    Re: Help with definite triple integral.

    Let

    $\displaystyle I=\int _{-2}^{-1}\int _{-2}^{-1}\int _{-2}^{-1}\frac{x^2}{x^2+y^2+z^2}dxdydz$

    $\displaystyle J=\int _{-2}^{-1}\int _{-2}^{-1}\int _{-2}^{-1}\frac{y^2}{x^2+y^2+z^2}dydxdz$

    $\displaystyle K=\int _{-2}^{-1}\int _{-2}^{-1}\int _{-2}^{-1}\frac{z^2}{x^2+y^2+z^2}dzdydx$

    now, $I=J=K$
    Last edited by Idea; Jun 18th 2019 at 11:07 PM.
    Thanks from topsquark
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