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Q10E

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Essential Calculus: Early Transcendentals

Found in: Page 734

Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

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Short Answer

Write the equations in cylindrical coordinates.
a. \({\rm{3x + 2y + z = 6}}\)
b. \({\rm{ - }}{{\rm{x}}^{\rm{2}}}{\rm{ - }}{{\rm{y}}^{\rm{2}}}{\rm{ + }}{{\rm{z}}^{^{\rm{2}}}}{\rm{ = 1}}\)

The equations in cylindrical coordinates \({\rm{z = 6 - 3rcos\theta - 2rsin\theta }}\)
The equations in cylindrical coordinates \({\rm{ - }}{{\rm{r}}^{\rm{2}}}{\rm{ + }}{{\rm{z}}^{\rm{2}}}{\rm{ = 1}}\)

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Concept Introduction

A multiple integral is a definite integral of a function of many real variables in mathematics (particularly multivariable calculus), such as f(x, y) or f(x, y) (x, y, z). Integrals of a two-variable function over a region

Step 2:Explanation of the solution

a.

To determine the cylindrical coordinates, apply the formulas below:

Remember that \({\rm{x = rcos}}\theta \) and \({\rm{y = rsin\theta }}\) in cylindrical coordinates. Because \({\rm{z = z}}\), we may substitute for x and\(\;y\) to get the following equation:

\({\rm{z = 6 - 3rcos\theta - 2rsin\theta }}\)

Step 3: Explanation of the solution

b.

Consider the given integral and simplify

In cylindrical coordinates, we employ the knowledge that \({\rm{x = rcos\theta }}\) and \({\rm{y = rsin\theta }}\) once more.

When we plug these into the equation, we get:

The equations in cylindrical coordinates \({\rm{ - }}{{\rm{r}}^{\rm{2}}}{\rm{ + }}{{\rm{z}}^{\rm{2}}}{\rm{ = 1}}\)

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