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Essential Calculus: Early Transcendentals
Found in: Page 734
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

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. \({\rm{ - }}{{\rm{x}}^{\rm{2}}}{\rm{ - }}{{\rm{y}}^{\rm{2}}}{\rm{ + }}{{\rm{z}}^{^{\rm{2}}}}{\rm{ = 1}}\)

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

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


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


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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