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Thread: Triple integrals

  1. #1
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    Question Triple integrals

    1. The problem statement, all variables and given/known data

    I have this question about triple integrals and spherical coordinates





    2. Relevant equations

    y = $\displaystyle \rho$ sin $\displaystyle \varphi$ sin $\displaystyle \theta$
    x = $\displaystyle \rho$ sin $\displaystyle \varphi$ cos $\displaystyle \theta$
    z = $\displaystyle \rho$ cos $\displaystyle \varphi$
    $\displaystyle \rho$^2 = z^2 + y^2 + x^2


    Thus I need to find the limits of integration for $\displaystyle \rho$ $\displaystyle \theta$ and $\displaystyle \varphi$

    3. The attempt at a solution

    I used the limits for the z to obtain z^2.
    Thus, z^2 + x^2+y^2 = 4
    Using the identity for $\displaystyle \rho$^2 = z^2 + y^2 + x^2 then $\displaystyle \rho$^2 = 4
    which gives me a value of $\displaystyle \rho$ = 2.

    To get $\displaystyle \theta$ I graphed the x limits of the integral. Since x = $\displaystyle \sqrt{4-y^2}$ then x^2 + y ^2 =4. Therefore it is a circle of radius 2. Thus I assumed that $\displaystyle \theta$ goes from 0 to 2$\displaystyle \pi$.
    Now my problem is to find the limits for $\displaystyle \varphi$ which I don't know how to get.

    Any ideas on how to solve for $\displaystyle \varphi$ and also, can someone double check that the other limits of integration are correct?

    Since z^2^ +y^2 + x^2 = 4 is a sphere and spheres have a $\displaystyle \phi$ from 0 to $\displaystyle \pi$. Can I use those limits for $\displaystyle \phi$. Can anybody double check that my limits of integration are correct?


    Thank you!
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  2. #2
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    Quote Originally Posted by jualin View Post
    1. The problem statement, all variables and given/known data

    I have this question about triple integrals and spherical coordinates





    2. Relevant equations

    y = $\displaystyle \rho$ sin $\displaystyle \varphi$ sin $\displaystyle \theta$
    x = $\displaystyle \rho$ sin $\displaystyle \varphi$ cos $\displaystyle \theta$
    z = $\displaystyle \rho$ cos $\displaystyle \varphi$
    $\displaystyle \rho$^2 = z^2 + y^2 + x^2


    Thus I need to find the limits of integration for $\displaystyle \rho$ $\displaystyle \theta$ and $\displaystyle \varphi$

    3. The attempt at a solution

    I used the limits for the z to obtain z^2.
    Thus, z^2 + x^2+y^2 = 4
    Using the identity for $\displaystyle \rho$^2 = z^2 + y^2 + x^2 then $\displaystyle \rho$^2 = 4
    which gives me a value of $\displaystyle \rho$ = 2.

    To get $\displaystyle \theta$ I graphed the x limits of the integral. Since x = $\displaystyle \sqrt{4-y^2}$ then x^2 + y ^2 =4. Therefore it is a circle of radius 2. Thus I assumed that $\displaystyle \theta$ goes from 0 to 2$\displaystyle \pi$.
    Now my problem is to find the limits for $\displaystyle \varphi$ which I don't know how to get.

    Any ideas on how to solve for $\displaystyle \varphi$ and also, can someone double check that the other limits of integration are correct?

    Since z^2^ +y^2 + x^2 = 4 is a sphere and spheres have a $\displaystyle \phi$ from 0 to $\displaystyle \pi$. Can I use those limits for $\displaystyle \phi$. Can anybody double check that my limits of integration are correct?

    Thank you!

    I tried 4 times to open the attached imageshack but it won't. Sorry.

    Tonio
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  3. #3
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    re:

    Here is the link again. Imageshack - 81255254.

    Or maybe this one will work
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