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Math Help - Home work check please - Triple integrals and centre of gravity

  1. #1
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    Home work check please - Triple integrals and centre of gravity

    I need to find the volume of region bounded above by sphere x^2 +y^2 +z^2=4 and below by the inverted cone z=-\sqrt{x^2 +y^2} , -\sqrt{2}\leqslant z \leqslant   0 and also find the centre of gravity with density equalling constant.



    I set up my bound and triple integral as


    \int_{0}^{(3\pi /4)}\int_{0}^{(2\pi)}\int _{0}^{\2} r^2\sin \varphi drd\theta d\varphi

    and after solving i got approx 28.60

    using symetry, the centre of gravity is on the z axis and using the formula for centre of mass:

    1/m\iiint z dV

    and subsituting above result and

    z = r\cos\varphi

    i obtained
    \iiint r\cos \varphi  r^2\sin \varphi drd\theta d\varphi

    solving with bounds stated in first part i got
    \bar{z} =  0.22

    does this look correct? thanks for any help.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by olski1 View Post
    I need to find the volume of region bounded above by sphere x^2 +y^2 +z^2=4 and below by the inverted cone z=-\sqrt{x^2 +y^2} , -\sqrt{2}\leqslant z \leqslant   0 and also find the centre of gravity with density equalling constant.



    I set up my bound and triple integral as


    \int_{0}^{(3\pi /4)}\int_{0}^{(2\pi)}\int _{0}^{\2} r^2\sin \varphi drd\theta d\varphi

    and after solving i got approx 28.60

    using symetry, the centre of gravity is on the z axis and using the formula for centre of mass:

    1/m\iiint z dV

    and subsituting above result and

    z = r\cos\varphi

    i obtained
    \iiint r\cos \varphi  r^2\sin \varphi drd\theta d\varphi

    solving with bounds stated in first part i got
    \bar{z} =  0.22

    does this look correct? thanks for any help.
    Without going through your working we can still check the numerical answers: Crude Monte-Carlo gives results close to yours, so yes they look OK

    (28.577 +/- 0.085 and 0.2186 +/- 0.0023 both 2 sigma error estimates)

    CB
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  3. #3
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    Thanks for your verification.

    Worried now that people in my course will just copy my working! any way i can hide it?
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  4. #4
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    Quote Originally Posted by olski1 View Post
    Thanks for your verification.

    Worried now that people in my course will just copy my working! any way i can hide it?
    See rule #7.

    Students who need to copy your working (assuming they see it) will hardly have sufficient grasp of the material to pass the exam. (And besides, if you are concerned about it being copied, then it seems likely that it forms part of a graded assignment, in which case we have unwittingly given you unfair help ....) You can always advise your instructor of this thread and suggest s/he check all homework against what you posted ....
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  5. #5
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    Quote Originally Posted by mr fantastic View Post
    See rule #7.

    Students who need to copy your working (assuming they see it) will hardly have sufficient grasp of the material to pass the exam. (And besides, if you are concerned about it being copied, then it seems likely that it forms part of a graded assignment, in which case we have unwittingly given you unfair help ....) You can always advise your instructor of this thread and suggest s/he check all homework against what you posted ....
    Thread closed.
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