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Thread: spherical coordinates

  1. #1
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    spherical coordinates

    Use spherical coordinates to show that $\displaystyle \int\int\int_{E} z dV = \frac{15\pi}{16}$, where $\displaystyle E$ lies between the spheres $\displaystyle x^2 + y^2 + z^2 = 1$ and $\displaystyle x^2 + y^2 + z^2 = 4$ in the first octant (i.e. $\displaystyle x, y, z \geq 0$).
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by wik_chick88 View Post
    Use spherical coordinates to show that $\displaystyle \int\int\int_{E} z dV = \frac{15\pi}{16}$, where $\displaystyle E$ lies between the spheres $\displaystyle x^2 + y^2 + z^2 = 1$ and $\displaystyle x^2 + y^2 + z^2 = 4$ in the first octant (i.e. $\displaystyle x, y, z \geq 0$).
    $\displaystyle z=\rho\cos\phi$
    $\displaystyle \rho^2 = x^2+y^2+z^2$
    $\displaystyle dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta$

    So the bounds are:

    $\displaystyle \rho: 1\to 2$
    $\displaystyle \phi: 0\to\pi/2$
    $\displaystyle \theta: 0\to\pi/2$

    This makes the integrand:

    $\displaystyle \rho\cos\phi*\rho^2\sin\phi\,d\rho\,d\phi\,d\theta = \rho^3\sin(\phi)\cos(\phi)\,d\rho\,d\phi\,d\theta$

    The integral is:

    $\displaystyle \int_0^{\pi/2}\int_0^{\pi/2}\int_1^2 \rho^3\sin(\phi)\cos(\phi)\,d\rho\,d\phi\,d\theta$

    Here's the evaluation of the integral, if you need it:
    Spoiler:

    Evaluating the innermost integral gives you $\displaystyle \left(\frac{\rho^4}{4}\sin(\phi)\cos(\phi)\right)\ bigg|_1^2 = \frac{15}{4}\sin(\phi)\cos(\phi)$

    Noticing that $\displaystyle \sin(\phi)\cos(\phi)=\frac{1}{2}\sin(2\phi)$, our integral is now:

    $\displaystyle \frac{15}{8}\int_0^{\pi/2}\int_0^{\pi/2}\sin(2\phi)\,d\phi\,d\theta$

    Evaluating the next integral, we have $\displaystyle \left(-\frac{1}{2}\cos(2\phi)\right)\bigg|_0^{\pi/2} = -\frac{1}{2}(-1-1) = 1$

    Now the integral is $\displaystyle \frac{15}{8}\int_0^{\pi/2}\,d\theta = \left(\frac{15}{8}\theta\right)\bigg|_0^{\pi/2} = \frac{15\pi}{16}$

    So the integral is $\displaystyle \frac{15\pi}{16}.$
    Last edited by redsoxfan325; Apr 23rd 2009 at 07:40 AM.
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