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Math Help - Triple Integral Using Spherical Coordinates

  1. #1
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    Triple Integral Using Spherical Coordinates

    Hi Everyone,
    I'm having a little problem with the following question that I was hoping someone could help me with.

    Basically I need to find the triple integral of (9 - x^2 -y^2) over a region, H, using spherical coordinates where H is the solid hemisphere
    x^2 + y^2 + z^2 <= 9,

    and z>=0.

    Now in terms of actually performing the triple integration, I have no problems, it is getting the spherical coordinates that is really bugging me.

    H should be in the following form

    H = {p,θ, φ| a<= 0 <= b, α<= θ<= β, c<=φ<=d}

    Now I know how to find p(really a and b). In this case a = 0 and b = 3.

    But I don't know how to find c, d, α, and β. I would really need a step-by-step guide (something like "Spherical Coordinates for dummies" )

    If anyone could let me know how to do this I would be most grateful.

    Also, if someone could provide me with another example of the process just to ensure that my grasp on the method is secure, then that would just be gravy.

    Eg.
    E is the area that is to be integrated over, where E lies between the spheres
    x^2 +y^2+z^2 = 1 and x^2 +y^2 +z^2 =4.
    (here a = 1, b=2)


    Thanks in advance to anyone who helps.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by woody198403 View Post
    Hi Everyone,
    I'm having a little problem with the following question that I was hoping someone could help me with.

    Basically I need to find the triple integral of (9 - x^2 -y^2) over a region, H, using spherical coordinates where H is the solid hemisphere
    x^2 + y^2 + z^2 <= 9,

    and z>=0.

    Now in terms of actually performing the triple integration, I have no problems, it is getting the spherical coordinates that is really bugging me.

    H should be in the following form

    H = {p,θ, φ| a<= 0 <= b, α<= θ<= β, c<=φ<=d}

    Now I know how to find p(really a and b). In this case a = 0 and b = 3.

    But I don't know how to find c, d, α, and β. I would really need a step-by-step guide (something like "Spherical Coordinates for dummies" )

    If anyone could let me know how to do this I would be most grateful.

    Also, if someone could provide me with another example of the process just to ensure that my grasp on the method is secure, then that would just be gravy.

    Eg.
    E is the area that is to be integrated over, where E lies between the spheres
    x^2 +y^2+z^2 = 1 and x^2 +y^2 +z^2 =4.
    (here a = 1, b=2)


    Thanks in advance to anyone who helps.
    As you correctly stated, the region projected onto the xy plane (z=0) gives you the limits 0\leq\varrho\leq3

    In this case, \varphi can only be the angle between the positive z and the xy plane [since z\geq0]. Thus, 0\leq\varphi\leq\frac{\pi}{2}

    So now we have two sets of limits.

    Now, lets find the limits for \vartheta.

    Since the region is a complete circle, \vartheta must be 2\pi. This is because 1 full revolution around a circle is 2\pi radians.

    Thus, we see that our integral is \int_0^{2\pi}\int_0^{\frac{\pi}{2}}\int_0^3\varrho  ^2\sin\varphi\,d\varrho\,d\varphi\,d\vartheta

    Does this somewhat make sense?

    --Chris
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  3. #3
    Super Member wingless's Avatar
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    Quote Originally Posted by Chris L T521 View Post
    Thus, we see that our integral is \int_0^{2\pi}\int_0^{\frac{\pi}{2}}\int_0^3\varrho  ^2\sin\varphi\,d\varrho\,d\varphi\,d\vartheta
    Chris, I think you forgot to put the function to be integrated in the region ;p ...

    Quote Originally Posted by woody198403 View Post
    Basically I need to find the triple integral of (9 - x^2 -y^2) over a region, H, using spherical coordinates where H is the solid hemisphere
    x^2 + y^2 + z^2 <= 9,
    But it should be no problem after finding the integration limits.

    Use these to get your function in spherical coordinates:
    x = r \cos \vartheta \sin \varphi
    y = r \sin \vartheta \sin \varphi
    z = r \cos \varphi
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by wingless View Post
    Chris, I think you forgot to put the function to be integrated in the region ;p ...



    But it should be no problem after finding the integration limits.

    Use these to get your function in spherical coordinates:
    x = r \cos \vartheta \sin \varphi
    y = r \sin \vartheta \sin \varphi
    z = r \cos \varphi
    I overlooked something... >_>

    Thanks for catching that!

    --Chris
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  5. #5
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    Exclamation

    Awesome, thanks so much. However, there is one other thing with this question that I just wanted to check which is whether this is correct?

    \int_0^{2\pi}\int_0^{\pi/2}\int_0^3 (9 - (p^2 sin^2 φ cos^2 θ) - (p^2 sin^2 φ sin^2 θ)) dp dφ dθ??

    im sure that "- (p^2 sin^2 φ cos^2 θ) - (p^2 sin^2 φ sin^2 θ))" is correct but im uncertain about what do do with the 9 here (only because im working with spherical coordinates).
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  6. #6
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    I was thinking that it could be

    \int_0^{2\pi}\int_0^{\pi/2}\int_0^3 ((p^2 cos^2 φ) - (p^2 sin^2 φ cos^2 θ) - (p^2 sin^2 φ sin^2 θ)) dp dφ dθ


    because if you look at x^2 + y^2 + z^2 <= 9, and let x=y=0, z= + or - 3, so z^2 = 9

    and

    z = p cos φ; so
    z^2 = p^2 cos^2 φ


    which I believe comes to
    \int_0^{2\pi}\int_0^{\pi/2}\int_0^3 p^2(cos 2φ) dp dφ dθ??


    am I way off here, or is this correct? Thanks.
    Last edited by woody198403; August 30th 2008 at 10:41 PM.
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