Calc 3

The region W is the cone shown below.
The angle at the vertex is 2π/3, and the top is flat and at a height of 5/sqrt(3). Write the limits of integration for ∫WdV in the following coordinates (do not reduce the domain of integration by taking advantage of symmetry):
(a) Cartesian:
(b) Cylindrical:
(c) Spherical:

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  1. The cone has a top radius of 5.
    In Cartesian coordinates, z^2 = (x^2+y^2)/6
    ∫W dV = ∫[-5,5]∫[-√(25-x^2),√(25-x^2) ∫[0,√((x^2+y^2)/6)] dz dy dz
    In cylindrical coordinates,
    ∫[0,2π] ∫[0,5] ∫[0,r/√3] r dz dr dØ
    In spherical coordinates, don't recall the transformations right off, but I'm sure your text has examples. I'll have to think on it a bit.

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