Find the volume of the part of the ball \rho \leq 16 that lies between the cones \theta = \pi 6...

Question:

Find the volume of the part of the ball {eq}\displaystyle \rho \leq 16 {/eq} that lies between the cones {eq}\displaystyle \theta =\frac{\pi }{6} {/eq} and {eq}\displaystyle \phi =\frac{\pi }{3}. {/eq}

Volume of the Region:

Note that the given equations are in spherical coordinate thus we use the triple integrals formula in spherical coordinates for the volume which is {eq}V(E) =\iiint\limits_{E}f\left ( x,y,z \right )dV=\int_{\theta _{1} }^{\theta _{2} }\int_{\phi _{1}}^{\phi _{2} }\int_{\rho _{1}}^{\rho _{1}}\rho ^{2}\sin \phi f\left ( \rho \sin \phi \cos \theta ,\rho \sin \phi \sin \theta ,\rho \cos \phi \right )d\rho d\theta d\phi {/eq} and the conversion formulas are {eq}x=\rho \sin \phi \cos \theta ,\:y=\rho \sin \phi \sin \theta ,\:z=\rho \cos \phi {/eq}

Answer and Explanation:

Based on the given equations the limits for integrations are the following,

{eq}0\leq \theta \leq 2\pi ,\:\frac{\pi }{6}\leq \phi \leq \frac{\pi }{3},\:0\leq \rho \leq 16 {/eq}

The volume of the region is,

{eq}V=\int_{0}^{2\pi }\int_{\frac{\pi }{6}}^{\frac{\pi }{3}}\int_{0}^{16}\rho ^{2}\sin \theta d\rho d\phi d\theta {/eq}

Integrate with respect to {eq}\rho {/eq}

{eq}V=\int_{0}^{2\pi }\int_{\frac{\pi }{6}}^{\frac{\pi }{3}}\left [ \frac{1}{3}\rho ^{3} \right ]^{16}_{0}\sin \theta d\phi d\theta {/eq}

{eq}V=\int_{0}^{2\pi }\int_{\frac{\pi }{6}}^{\frac{\pi }{3}}\frac{4096}{3}\sin \theta d\phi d\theta {/eq}

Integrate with respect to {eq}\phi {/eq}

{eq}V=\int_{0}^{2\pi }\frac{4096}{3}\left [ -\cos \left(\phi \right) \right ]^{\frac{\pi }{3}}_{\frac{\pi }{6}} d\theta {/eq}

{eq}V=\int_{0}^{2\pi }\left [ \frac{2048\left(-1+\sqrt{3}\right)}{3} \right ] d\theta {/eq}

Integrate with respect to {eq}\theta {/eq}

{eq}V=\left [ \frac{2048\left(-1+\sqrt{3}\right)}{3} \right ] \left [ \theta \right ]^{2\pi }_{0} {/eq}

{eq}V=\frac{4096\pi \left(-1+\sqrt{3}\right)}{3} {/eq}

{eq}V=\:3140.0 {/eq}


Learn more about this topic:

Loading...
Volumes of Shapes: Definition & Examples

from GMAT Prep: Tutoring Solution

Chapter 11 / Lesson 9
45K

Related to this Question

Explore our homework questions and answers library