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Substituting spherical coordinates to evaluate an integral

  1. Dec 1, 2015 #1
    I have to evaluate

    $$\int^1_{-1} \int^{ \sqrt {1-x^2}}_{ - \sqrt {1-x^2}} \int^1_{-\sqrt{x^2+y^2}}dzdydx$$

    using spherical coordinates.

    This is what I have come up with

    $$\int^1_{0} \int^{ 2\pi}_0 \int^{3\pi/4}_{0}r^2\sin\theta d\phi d\theta dr$$

    by a combination of sketching and substituting spherical coordinates.

    After evaluating I obtain this integral to equal 3.57.

    where as the first one evaluates to 5.236.

    These are so difficult :(
  2. jcsd
  3. Dec 1, 2015 #2


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    Homework Helper

    I am not convinced you did that right.
    The volume of integration appears to be a sphere for z<0 and a cylinder for z>0. Your spherical integral doesn't look like that.

    Are you allowed to use spherical integral for the lower half and cylindrical integral for the upper half? Or maybe just geometry...1/2 volume of unit sphere + volume of unit cylinder = 5.236.
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