Sketch the region E that lies inside the cone \phi = \frac{\pi}{3} and inside the sphere \rho = 4...

Question:

Sketch the region E that lies inside the cone {eq}\phi = \frac{\pi}{3} {/eq} and inside the sphere {eq}\rho = 4 \cos (\phi) {/eq}. Then find the volume of E.

Triple Integrals in Spherical Coordinates

The volume of a solid region E given in spherical coordinates,

{eq}\displaystyle x=\rho\cos \theta \sin \varphi, y=\rho\sin \theta\sin \varphi, z=\rho \cos \varphi, {/eq} as {eq}\displaystyle E=\{(\rho,\theta, \varphi)|\, \theta_1\leq \theta\leq \theta_2, f_1(\theta)\leq \varphi\leq f_2( \theta),g_1(\theta, \varphi)\leq \rho\leq g_2(\theta, \varphi)\} {/eq}

has the volume calculated by the following triple integral

{eq}\displaystyle \iiint_{E}\ dV= \int_{\theta_1}^{\theta_2}\int_{ f_1(\theta)}^{ f_2( \theta)}\int_{g_1(\theta, \varphi)}^{g_2(\theta, \varphi)} \rho^2 \sin \varphi\, d\rho d\varphi d\theta . {/eq}

Answer and Explanation: 1

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To find the volume of the region {eq}\displaystyle E {/eq} inside the cone {eq}\displaystyle \varphi=\frac{\pi}{3}, {/eq} and inside the sphere ...

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Cylindrical & Spherical Coordinates: Definition, Equations & Examples

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Chapter 13 / Lesson 10
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In this lesson, we introduce two coordinate systems that are useful alternatives to Cartesian coordinates in three dimensions. Both cylindrical and spherical coordinates use angles to specify the locations of points, a feature they share with 2-D polar coordinates.


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