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Math Help - 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 = \rho sin \varphi sin \theta
    x = \rho sin \varphi cos \theta
    z = \rho cos \varphi
    \rho^2 = z^2 + y^2 + x^2


    Thus I need to find the limits of integration for \rho \theta and \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 \rho^2 = z^2 + y^2 + x^2 then \rho^2 = 4
    which gives me a value of \rho = 2.

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

    Any ideas on how to solve for \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 \phi from 0 to \pi. Can I use those limits for \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 = \rho sin \varphi sin \theta
    x = \rho sin \varphi cos \theta
    z = \rho cos \varphi
    \rho^2 = z^2 + y^2 + x^2


    Thus I need to find the limits of integration for \rho \theta and \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 \rho^2 = z^2 + y^2 + x^2 then \rho^2 = 4
    which gives me a value of \rho = 2.

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

    Any ideas on how to solve for \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 \phi from 0 to \pi. Can I use those limits for \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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    Nov 2009
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    re:

    Here is the link again. Imageshack - 81255254.

    Or maybe this one will work
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