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Triple integrals

  1. Nov 21, 2009 #1
    1. The problem statement, all variables and given/known data

    I have this question about triple integrals and spherical coordinates

    http://img405.imageshack.us/img405/9343/81255254.th.jpg [Broken]



    2. Relevant equations

    y = [tex]\rho[/tex] sin [tex]\varphi[/tex] sin [tex]\theta[/tex]
    x = [tex]\rho[/tex] sin [tex]\varphi[/tex] cos [tex]\theta[/tex]
    z = [tex]\rho[/tex] cos [tex]\varphi[/tex]
    [tex]\rho[/tex]2 = z2 + y2 + x2

    This is the way
    http://tutorial.math.lamar.edu/Classes/CalcIII/TISphericalCoords_files/eq0007MP.gif" [Broken]

    Thus I need to find the limits of integration for [tex]\rho[/tex] [tex]\theta[/tex] and [tex]\varphi[/tex]

    3. The attempt at a solution

    I used the limits for the z to obtain z2.
    Thus, z2 + x2 +y2 = 4
    Using the identity for [tex]\rho[/tex]2 = z2 + y2 + x2 then [tex]\rho[/tex]2 = 4
    which gives me a value of [tex]\rho[/tex] = 2.

    To get [tex]\theta[/tex] I graphed the x limits of the integral. Since x = [tex]\sqrt{4-y2}[/tex] then x2 + y 2 =4. Therefore it is a circle of radius 2. Thus I assumed that [tex]\theta[/tex] goes from 0 to 2[tex]\pi[/tex].
    Now my problem is to find the limits for [tex]\varphi[/tex] which I don't know how to get.

    Any ideas on how to solve for [tex]\varphi[/tex] and also, can someone double check that the other limits of integration are correct?

    Thank you!
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Nov 21, 2009 #2
    How do you derive the spherical coordinates? You can find the ranges of [tex]\phi[/tex] in the definition of spherical coordinates, so study your book again! And the ranges for [tex]\rho[/tex] and [tex]\theta[/tex] are correct.
     
  4. Nov 21, 2009 #3
    Can I use the limits of y to get [tex]\phi[/tex]. For instance since y = 4 then can I say
    [tex]\rho[/tex] sin [tex]\phi[/tex] sin [tex]\theta[/tex] = 4 so sin [tex]\phi[/tex] = [tex]\rho[/tex] / sin [tex]\theta[/tex]

    Now I am stuck there. Do I plug in a value for [tex]\rho[/tex] and [tex]\theta[/tex]. For instance 2 for [tex]\rho[/tex] and 2pi for [tex]\theta[/tex]. That would give me an undefined answer and sin [tex]\phi[/tex] is always defined. Where do I go from here?
    Thank you for the quick response
     
  5. Nov 21, 2009 #4
    Or since z2 +y2 + x2 = 4 is a sphere and spheres have a [tex]\phi[/tex] from 0 to [tex]\pi[/tex]. Can anybody double check that my limits of integration are correct?

    Thank you
     
    Last edited: Nov 21, 2009
  6. Nov 21, 2009 #5
    Yes your bounderies are now correct and I'm sure about that, because I have a similar example in my book with answer.

    [tex]\rho[/tex] is between 0 to 2 and [tex]\theta[/tex] is between 0 to [tex]2\pi[/tex] and [tex]\phi[/tex] is between 0 to [tex]\pi[/tex]
     
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