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Thread: 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 $\displaystyle \int_{0}^{2}\int_{0}^{\sqrt{4-y^2}}\int_{\sqrt{x^2+y^2}}^{\sqrt{8-x^2-y^2}}\!z\ \mathrm{d}z\ \mathrm{d}x\ \mathrm{d}y$. 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 $\displaystyle xy$-plane is the portion of the circle $\displaystyle x^2+y^2=4$ on $\displaystyle 0\le x\le2,\ y\ge0$.

    The bounds on $\displaystyle z$ correspond to

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

    $\displaystyle x^2+y^2=4$.
    Below $\displaystyle z=2$ (where the bounds on $\displaystyle z$ intersect), I believe that the cone and cylinder, $\displaystyle x^2+y^2=4$, 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 $\displaystyle x\ge0$, $\displaystyle y\ge0$ and $\displaystyle 0\le z\le2$? 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,

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

    $\displaystyle \int_{0}^{\frac{\pi}{2}}\int_{0}^{\frac{\pi}{2}} \int_{0}^{2\sqrt{2}}\!\rho\ \cos\phi\ \rho^2 \sin \phi\ \mathrm{d}\rho\ \mathrm{d}\theta\ \mathrm{d}\phi=4\pi$
    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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