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Math Help - computing area

  1. #1
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    computing area

    Hi,
    I have this problem:
    the surfaces r=2 and 4, \theta=30 degrees and 50 degrees, \phi=20 degrees and 60 degrees identify a closed surface.
    1- find the enclosed volume.
    2- Find the total area of the enclosed surface. ( I think it is a typo from the teacher. It is volume not surface)

    The first question is straigth forward
    For the secon question I have some issues.
    Do I need to take take each element of surface (in spherical coordinates)
    ,dS1= r^2 \sin\theta d\theta d\phi
    dS2= r dr d\phi
    dS3= \sin\theta r dr d\phi

    then integrate the according the limits and the total are should be
    S= S1+S2+S3
    Is my reasonnig correct?
    Thank you
    B
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  2. #2
    Eater of Worlds
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    It would appear it is a region enclosed by two concentric spheres.

    {\rho}=2 \;\ and \;\ {\rho}=4 are two spheres of respective radius 2 and 4.

    \Large\int_{\frac{\pi}{6}}^{\frac{5{\pi}}{18}}\int  _{\frac{\pi}{9}}^{\frac{\pi}{3}}\int_{2}^{4}{\rho}  ^{2}sin({\phi})d{\rho}d{\phi}d{\theta}
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  3. #3
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    Galactus ,
    thank you but I think that this is the question for the first question which is compute the volume enclosed.
    The second question is the one that bothers me.
    2- Find the total area of the enclosed Volume.
    B.
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  4. #4
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    Quote Originally Posted by braddy View Post
    Galactus ,
    thank you but I think that this is the question for the first question which is compute the volume enclosed.
    The second question is the one that bothers me.


    B.
    Your reasoning is correct, except that the differential of the vertical arc is r*sin(phi)*d(phi).
    Not r*sin(theta)*d(theta).

    So your d(S1), for the concave/convex surfaces at r=2 and r=4 each, should be
    [r*sin(phi)*d(phi)]*[r*d(theta)]
    = (r^2)(sin(phi))(d(phi))(d(theta))

    Yours is (r^2)(sin(theta))(d(theta))(d(phi)).

    [Well, unless, of course, your vertical angle is theta, and your horizontal angle is phi. If that is the case, then your d(S1) is correct. It's just not the conventional way of doing it.]

    -------------
    Then, d(S2), for the "horizontal" walls, should be (r*d(theta))(dr) = r*dr*d(theta)

    ------------
    And, d(S3), for the "vertical" walls, should be (r*sin(phi)*d(phi))(dr) = r*sin(phi)*d(phi)*dr.
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