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Thread: [SOLVED] triple integral issue

  1. #1
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    [SOLVED] triple integral issue

    I need some help with a triple integral problem that I keep getting wrong in the last step somehow.

    the problem is:

    Evaluate $\displaystyle \int\int\int$ on E xyz dV where E lies between the spheres $\displaystyle \rho$=2 and $\displaystyle \rho$=4 and above the cone $\displaystyle \phi$=$\displaystyle \pi$/3.

    There work I've done so far is set up the integral integrated it and got:

    1/4 sin^4$\displaystyle \phi$ from 0 to $\displaystyle \pi$ * 1/2sin^2$\displaystyle \theta$ from 0 to 2$\displaystyle \pi$ * 1/6 $\displaystyle \rho$^6 from 2 to 4.

    I get that the middle term should be zero, but for some reason my prof gives the answer to be
    (1562$\displaystyle \pi$)/15
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  2. #2
    Eater of Worlds
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    That is a large value your professor got. I am not so sure he/she is correct.
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  3. #3
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    That's what I was thinking as I am pretty sure the answer should be zero, but I thought that it wouldn't hurt to check.
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  4. #4
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    Consider symmetry. The region of integration is rotationally symmetric about the z axis, and so if we rotate the function being integrated about the z axis, the integral should be unchanged. Now, let us rotate the integrand by 90 degrees about the z axis: we transform x and y to x' and y' via x'=-y and y'=x. Then the integrand is f(x',y',z)=xyz=(y')(-x')z=-x'y'z.
    Thus, we have $\displaystyle \iiint_{E}xyz\,dx\,dy\,dz=\iiint_{E'}-x'y'z\,dx'\,dy'\,dz$. But, if we rename the variables x' and y' as x and y, (and as E'=E), we have
    $\displaystyle \iiint_{E}xyz\,dx\,dy\,dz=-\iiint_{E}xyz\,dx\,dy\,dz$
    which means $\displaystyle \iiint_{E}xyz\,dx\,dy\,dz=0$.
    Thus, due to the symmetry, the integral is zero.
    [We can also look at this as the fact that xyz's dependence on the azimuthal angle $\displaystyle \theta$ is of the form $\displaystyle \sin(2\theta)$, and the region has no dependence on the azimuthal angle (the rotational symmetry), so the integral over $\displaystyle \theta$ will be 0]

    --Kevin C.
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  5. #5
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    Thanks everyone. I got an e-mail back from my professor and he said that it was a typo. Thanks again!
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