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Set up the integral to calculate the volume in cylindrical coordinates bounded by z=...

Question:

Set up the integral to calculate the volume in cylindrical coordinates bounded by

{eq}z= \sqrt{25-r^2}, \ z=4, \ r=4 {/eq}.

Cylindrical Coordinates, Volume of a Solid:

The volume of a solid region T, is given by the triple integral,

{eq}\displaystyle \iiint_T \; dV. {/eq}

When the solid is cylindrical in shape, we resort to cylindrical coordinates, which are a combination of the polar coordinates, and the z - coordinate. The cartesian coordinates are represented as,

{eq}x = r \cos \theta \\ y = r \sin \theta \\ z = z {/eq}

The differential volume element in cylindrical coordinates becomes,

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

The volume expression is of a small solid shape like the bite off of a pizza slice.

The above concept can be assimilated from the following figure:

Answer and Explanation: 1

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Observe the graph of the solid region bounded by {eq}z= \sqrt{25 - r^2}, \ z=4 {/eq}.

The region can be parametrized using cylindrical...

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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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