Use polar coordinates to compute the volume of the given solid. Inside the sphere x^2 + y^2 + z^2...

Question:

Use polar coordinates to compute the volume of the given solid.

Inside the sphere {eq}x^2 + y^2 + z^2 = 25 {/eq} and outside the cylinder {eq}x^2 + y^2 = 1 {/eq}.

Cylindrical Coordinates:

Since our region has a cylinder in it, it will be most convenient to use cylindrical coordinates, which are the usual polar coordinates with an extra dimension, {eq}z {/eq}, added in to manage depth. The relations between polar and Cartesian coordinates are

{eq}x = r \cos \theta {/eq}

{eq}y = r \sin \theta {/eq}

{eq}z = z {/eq}

{eq}r^2 = x^2+y^2 {/eq}

{eq}\theta = \tan^{-1} \frac{y}{x} {/eq}

{eq}dV = r \ dz \ dr \ d\theta {/eq}

Answer and Explanation: 1

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In cylindrical coordinates, the cylinder is simply {eq}r = 1 {/eq}. Since the radial variable is bounded below by the cylinder and above by the...

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Cylindrical & Spherical Coordinates: Definition, Equations & Examples

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Chapter 13 / Lesson 10
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Learn how to convert between Cartesian, cylindrical and spherical coordinates. Discover the utility of representing points in cylindrical and spherical coordinates.


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