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Tripple integral

  1. Jan 10, 2012 #1
    1. The problem statement, all variables and given/known data

    find the value of [tex]\int\int\int_D{e^{(x^2+y^2+z^2)^{3/2}/2}}dV[/tex] given [tex]1\leq{x^2+y^2+z^2}\leq{3}, z^2\geq{2}(x^2+y^2), 2x\leq{y}\leq3x[/tex]
    I am reviewing tripple integrals and am having a bit of difficulty determining the limits for each part. I have,
    [tex]arctan(2)\leq\theta\leq{arctan(3)}, 1\leq\rho\leq{\sqrt{3}}[/tex]

    but I cant seem to visualize what is happening with
    [tex]\phi[/tex]
    Any suggestions?
     
    Last edited: Jan 11, 2012
  2. jcsd
  3. Jan 10, 2012 #2

    Dick

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    Can you visualize what the surface z^2=2*(x^2+y^2) looks like?
     
  4. Jan 10, 2012 #3
    Kind of; paraboloid? If it is indeed a paraboloid, how does one determine the angle phi when the parabloid has a curved surface?
    [tex]\phi{_1}\leq\phi\leq\pi/2[/tex]
     
  5. Jan 10, 2012 #4

    Dick

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    z=2*(x^2+y^2) is a paraboloid. z^2=2*(x^2+y^2) isn't. Think a little harder. I'll give you a hint. Sketch z^2=2*x^2 in the x-z plane.
     
    Last edited: Jan 10, 2012
  6. Jan 10, 2012 #5
    It's a cone, I believe. Now, this cone is going to intersect the outer edge of the sphere, and therefore, plugging the equation for the cone into x^2+y^2+z^2=3, we get
    [tex]3x^2+3y^2=3\rightarrow{x^2+y^2=1}[/tex]
    Do I have the right idea so far?
     
  7. Jan 10, 2012 #6

    Dick

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    You have the right idea that it's a cone. Doesn't the cone determine the range of ϕ? Isn't that what you were asking about?
     
  8. Jan 11, 2012 #7
    Indeed, it does. However, is this not limited by the point that the cone intersects the outer edge of shell? This is why I combined the equations
    [tex]x^2+y^2+z^2=3, z^2=2(x^2+y^2)[/tex]
    This point of interesction can then be used to determine the angle from the xy-plane, which phi will merely be 90 less the angle measured.
     
  9. Jan 11, 2012 #8
    Here is the drawing:
    yes-4.jpg
    So [tex]\phi[/tex] will go from the angle that it intersects to pi/2
     
    Last edited: Jan 11, 2012
  10. Jan 11, 2012 #9

    Dick

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    Right, but I don't think there is any need to intersect z^2=2(x^2+y^2) with anything. ANY point on the cone that's above the x-y plane has the same value of phi.
     
  11. Jan 11, 2012 #10
    This is true. I got the right answer. Thanks
     
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