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Homework Help: Evaluate triple integral

  1. Nov 8, 2012 #1
    1. The problem statement, all variables and given/known data

    Evaluate triple integral

    z^2 dxdydz

    i) the part of the sphere x^2 + y^2 + z^2 = a^2 (first octant)
    ii)the complete interior of the sphere x^2 + y^2 + z^2 = a^2 (first octant)

    2. Relevant equations

    It is probably good idea to work in spherical coords.

    z = r*cosφ
    x = r*sinφ cosθ
    y = r*sinφ sinθ

    dxdydz = r^2 sinφ drdφdθ

    3. The attempt at a solution

    I'l start at part ii) because its the part I can do.
    Here the boundaries are:
    0 =< r < a
    0 =< φ < pi/2
    0 =< θ < pi/2

    the integration now becomes:

    (Int[r=0, a] r^4 dr )( Int[φ=0, pi/2] sinφcos^2 φ)( Int [θ=0, pi/2]) = r^5/30 * pi

    i) But for part i), I am confused. The integral should be evaluated only on the surface of the sphere. The radius a is constant in length, so how should r be defined?
    a < r < a, makes no sense.

    Need advice.
    Last edited: Nov 8, 2012
  2. jcsd
  3. Nov 8, 2012 #2


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    Are you sure the problem is quoted correctly?
  4. Nov 8, 2012 #3


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    Part i obviously means the interior of the sphere in the first octant. Otherwise it wouldn't be a triple integral.
  5. Nov 9, 2012 #4
    You are right, I understand what nonsense I was thinking about. The part i) is not quoted correctly. Thanks!
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