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Triple Integral Problem

  1. Nov 30, 2014 #1
    1. The problem statement, all variables and given/known data
    Evaluate ∫∫∫[W] xz dV, where W is the domain bounded by the elliptic cylinder (x^2)/4 + (y^2)/9 = 1 and the sphere x^2 + y^2 + z^2 = 16 in the first octant x> or = 0, y> or = 0, z> or = 0.

    2. Relevant equations
    First, I tried to find the bounds for z:
    z = 0 (because z is greater than or equal to zero) to z = sqrt(16 - x^2 - y^2).

    Then setting z = 0, I tried to find the x bounds:
    x = sqrt(4 - (4y^2)/9) to x = sqrt(16 - y^2).

    Finally with both x and z set to 0, I tried to find the y bounds:
    y = 3 to y = 4.

    3. The attempt at a solution
    3 to 4 sqrt(4 - (4y^2)/9) to sqrt(16 - y^2) 0 to sqrt(16 - x^2 - y^2) xz dzdxdy

    When I tried to solve this, it didn't work. I'm wondering if I have the bounds wrong.
     
  2. jcsd
  3. Nov 30, 2014 #2

    Zondrina

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    Homework Helper

    EDIT:

    You should find the intersection of the sphere and cylinder by setting them both to be zero and solving. This will give you an ellipsoid.

    Using the transformation ##x = \sqrt{20} u##, ##y = \sqrt{\frac{135}{8}} v##, and ##z = \sqrt{15}w## will transform the ellipsoid into a sphere of radius 1. Computing the Jacobian of this transformation will allow you to transform the original integral:

    $$\iiint_V xz \space dV = \iiint_{V'} (\sqrt{20} u)(\sqrt{15}w) \space |J| \space dV'$$

    I believe a change to spherical co-ordinates from here will clean up the limits and the integral.
     
    Last edited: Nov 30, 2014
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