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Multiple integral in a domain

  1. Sep 6, 2015 #1
    1. The problem statement, all variables and given/known data
    Hi, I have a problem with the following exercise.
    Let C={(x,y,z)∈ℝ3 : x2+y2+z2≤1, z≥√(3x2+8y2)} be a subset of ℝ3. Calculate ∫∫∫C z dxdydz.


    2. Relevant equations

    Spherical coordinates, given by x=ρsinΦcosθ, y=ρsinΦsinθ, z=ρcosΦ, and cylindrical coordinates which are x=ρcosα, y=ρsinα, z=z

    3. The attempt at a solution

    The problem is that, starting from the set C I believe that √(3x2+8y2)≤z≤√(1-x2-y2) so that I should integrate the "∫zdz part" between these two extremes. But when it comes to the conditions on x and y I don't know how to proceed further. I tried to use spherical coordinates so that the first condition becomes ρ2≤1 but then the second one becomes cos(Φ)≥√(3sin2(Φ)cos2(θ)+8sin2(Φ)sin2(θ)) which is worse than the original condition. I also tried to use cylindrical coordinates so that even though the first condition does not improve much the second one becomes z2≥3ρ2cos2(α)+8ρ2sin2(α), which is better than the one I obtained using spherical coordinates, but even so I didn't go very far. Obviously there is some idea I am missing, what is in your opinion the best way to go on?
     
    Last edited: Sep 6, 2015
  2. jcsd
  3. Sep 6, 2015 #2

    Zondrina

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    You are right about using spherical co-ordinates. Applying the transformation will yield ##0 \leq \rho \leq 1## as you have found already.

    The limits for ##\theta## can be found by using your imagination. Are you integrating over a full ##2 \pi## or only half of that? If you sketch the region, it will be clear what the limits for ##\theta## are.

    As for ##\phi##, you have the right idea, but you seem to have stopped mid-computation. Once you get here:

    $$\cos( \phi ) \geq \sqrt{3 \cos^2( \theta ) \sin^2( \phi ) + 8 \sin^2( \theta ) \sin^2( \phi )}$$

    Factor out ##\sin^2( \phi )## to obtain:

    $$\cot( \phi ) \geq \sqrt{3 \cos^2( \theta ) + 8 \sin^2( \theta )}$$
     
  4. Sep 6, 2015 #3

    Ray Vickson

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    Personally, I would not use cylindrcal or spherical coordinates; I would first just attempt to determine the ##(x,y)## region ##R_{xy}##, which is the region where ##\sqrt{1-x^2-y^2} \geq \sqrt{3x^2+8y^2}. ##
    Then I would do the integral
    [tex] \int \! \int_{R_{xy}} \left( \int_{\sqrt{3x^2+8y^2}}^{\sqrt{1-x^2-y^2}} z \, dz \right) \, dx dy .[/tex]
     
  5. Sep 6, 2015 #4

    Zondrina

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    This way is nice too because you can switch over to polar co-ordinates once you obtain an ##xy## integral.
     
  6. Sep 6, 2015 #5
    Thank you very much! I will try to use both the approaches. I didn't go further with the calculations of the second conditions using spherical coordinates because I didn't understand how to use that condition but now I understand that I needed your last passage, and I didn't really thought that I could switch to spherical coordinates midway through the integration. I will try to solve the integral and when I'm done I will post my solution here
     
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