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Converting to Spherical Coordinates then integrating? Am I doing this right?

  1. May 29, 2012 #1
    Converting to Spherical Coordinates....then integrating? Am I doing this right?

    1. The problem statement, all variables and given/known data
    Consider the integral ∫∫∫(x2z + y2z + z3) dz dy dx, where the left-most integral is from -2 to 2, the second -√(4-x2) to √(4-x2) and the right-most integral is from 2-√(4-x2-y2) to 2+√(4-x2-y2)


    2. Relevant equations
    x=ρsin(phi)cos(θ)
    y=ρsin(phi)sin(θ)
    z=ρcos(phi)
    x2+y2+z22


    3. The attempt at a solution
    So I need to convert to spherical coordinates. The first 2 integrals describe a region on the xy-plane that's a circle, centered at the origin, with a radius of 2. The distance from the origin I thought was from 0 to 4, as described the right-most integral from above. Then using the equations from above, I converted the integrand:

    ∫∫∫ρcos(phi)ρ2sin(phi) dρ d(phi) dθ

    Left-most integral: 0 to 2*pi
    Middle: 0 to pi/2
    Right: 0 to 4

    Final answer = 2144.67 (which does not feel right)

    I'm fairly certain I am correct for the most part, except the right most integral in the spherical-converted integral. Anyone care to check if I did everything correctly?
     
  2. jcsd
  3. May 29, 2012 #2

    SammyS

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    Re: Converting to Spherical Coordinates....then integrating? Am I doing this right?

    Hello emzee1. Welcome to PF !

    The volume element in spherical coordinates is [itex]dV=\rho^2\sin(\phi)\,d\rho\,d\phi\,d\theta\ .[/itex]

    The integrand: [itex]x^2z+y^2z+z^3=(x^2+y^2+z^2)z \ \to\ \rho^2\left(\rho\cos(\phi)\right)=\rho^3\cos(\phi)\ .[/itex]
     
  4. May 29, 2012 #3
    Re: Converting to Spherical Coordinates....then integrating? Am I doing this right?

    Thanks for that SammyS, I guess that was a mental mistake on my part. What about the limits for the integral for the spherical-integral? I don't think I did those correctly...
     
  5. May 29, 2012 #4

    SammyS

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    Re: Converting to Spherical Coordinates....then integrating? Am I doing this right?

    Look at the limits of integration for z.

    They describe a sphere of radius 2, centered at (x, y, z) = (0, 0, 2) .

    Write that equation in spherical coordinates.
     
  6. May 29, 2012 #5
    Re: Converting to Spherical Coordinates....then integrating? Am I doing this right?

    So I converted the equation of the sphere:

    x2+y2+(z-2)2 = 4

    to:

    ρ2-4ρcos(phi) = 0

    solving for ρ:

    ρ= 4cos(phi)

    so the integral limits, in terms of dρ: 0 to 4cos(phi) ?
    Then the limits of d(phi): 0 to pi/2
    And the limits of dθ: 0 to 2*pi?

    Does this sound correct?
     
  7. May 29, 2012 #6

    SammyS

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    Re: Converting to Spherical Coordinates....then integrating? Am I doing this right?

    That looks better --- correct.

    I'm notorious for overlooking details! LOL !
     
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