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}

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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 ...

Cylindrical & Spherical Coordinates: Definition, Equations & Examples

from

Chapter 13 / Lesson 10
19K

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.