Explanation Explanation Given 1. Description of the three dimensional region in Cartesian coordinates. The region is bounded below the paraboloid z = 6 − ( x 2 + y 2 ) and bounded above the plane z = 2. Formula 1. Equations which transform Cartesian coordinates into spherical polar coordinates x = r cos ( θ ) , y = r sin ( θ ) , and z with r ≥ 0 and 0 ≤ θ ≤ 2 π . Calculation a. Each plane parallel to the x − y plane of the form z = k intersect with the parabaloid in disc with radius 6 − k . Define the polar coordinates for this disc as x = r cos ( θ ) and y = r sin ( θ ) where θ varies from 0 to 2 π . Use the equation of the paraboloid to obtain the limits of r . We have z = 6 − ( r 2 cos 2 ( θ ) + r 2 sin 2 ( θ ) ) = 6 − r 2 . Therefore r varies from 0 to 6 − z . When x = y = 0 , then z = 6 . Therefore z varies from 2 to 6. Hence equation of the region in the cylindrical coordinates are given by x = r cos ( θ ) , y = r sin ( θ ) and z where 0 ≤ θ ≤ 2 π , 0 ≤ r ≤ 6 − z and 2 ≤ z ≤ 6 . b...

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Chapter 16.4, Problem 25E

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Equation of a region in the three dimensional space using cylindrical polar coordinates and then compute volume of this region.

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Chapter 16.4, Problem 24E

Chapter 16.4, Problem 26E

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Chapter 16 Solutions

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To find Equation of a region in the three dimensional space using cylindrical polar coordinates and then compute volume of this region.

Solution Summary: The author explains how to find Equation of a region in the three dimensional space using cylindrical polar coordinates and then compute volume of this region.

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To find Equation of a region in the three dimensional space using cylindrical polar coordinates and then compute volume of this region.

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Chapter 16 Solutions

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Equation of a region in the three dimensional space using cylindrical polar coordinates and then compute volume of this region.