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Math Help - Describing/sketching region of integration of triple integral

  1. #1
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    Describing/sketching region of integration of triple integral

    Hi there,

    I'm having great difficulty with the following problem:

    This question concerns the integral \int_{0}^{2}\int_{0}^{\sqrt{4-y^2}}\int_{\sqrt{x^2+y^2}}^{\sqrt{8-x^2-y^2}}\!z\ \mathrm{d}z\ \mathrm{d}x\ \mathrm{d}y. Sketch or describe in words the domain of integration. Rewrite the integral in both cylindrical and spherical coordinates. Which is easier to evaluate?
    Below is what I believe I have established so far...

    The projection of this integral's domain onto the xy-plane is the portion of the circle x^2+y^2=4 on 0\le x\le2,\ y\ge0.

    The bounds on z correspond to

    z^2=x^2+y^2 (cone) and x^2+y^2+z^2=8 (sphere).
    These bounds intersect at

    x^2+y^2=4.
    Below z=2 (where the bounds on z intersect), I believe that the cone and cylinder, x^2+y^2=4, are completely inside the sphere.

    Would it hence be correct to say that the region of integration is the solid lying between the cone and the cylinder, on x\ge0, y\ge0 and 0\le z\le2? I'm struggling to visualize this problem.

    When I attempt to move on, and evaluate the integral in cylindrical/spherical coordinates, my solutions differ by a factor of 2.

    That is, I evaluated this integral as,

    \int_{0}^{\frac{\pi}{2}}\int_{0}^{2}\int_{0}^{ \sqrt{8-r^2}}\!z\ r\ \mathrm{d}z\ \mathrm{d}r\ \mathrm{d}\theta=2\pi
    And,

    \int_{0}^{\frac{\pi}{2}}\int_{0}^{\frac{\pi}{2}} \int_{0}^{2\sqrt{2}}\!\rho\ \cos\phi\ \rho^2 \sin \phi\ \mathrm{d}\rho\ \mathrm{d}\theta\ \mathrm{d}\phi=4\pi
    Can you please help me to identify where I am going wrong?

    Thank you very much.
    Last edited by drokkin; May 26th 2013 at 06:49 PM.
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  2. #2
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    Re: Describing/sketching region of integration of triple integral

    Hey drokkin.

    Can you fix up your tex code? I can't really make out what the equations are.
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  3. #3
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    Re: Describing/sketching region of integration of triple integral

    Quote Originally Posted by chiro View Post
    Hey drokkin.

    Can you fix up your tex code? I can't really make out what the equations are.
    Sorry - hit the "Submit" button instead of the "Preview" button! All sorted now.
    Last edited by drokkin; May 26th 2013 at 06:34 PM.
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