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Use triple itegral.in cylindrical. coordinates to mass of a- find. sphere .with density S. te. ass and raolius 2 ceutered at 12,0,0).

Use triple itegral.in cylindrical. coordinates to mass of a- find. sphere .with density S. te. ass and raolius 2 ceutered at 12,0,0).

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Title:** Calculating the Mass of a Sphere Using Triple Integrals in Cylindrical Coordinates **Objective:** Learn how to use triple integrals in cylindrical coordinates to determine the mass of a given sphere. **Problem Statement:** Use a triple integral in cylindrical coordinates to find the mass of a sphere with a density function \( \delta = z \) and a radius of 2, centered at \( (2, 0, 0) \). **Solution Steps:** 1. **Convert to Cylindrical Coordinates:** - Consider the transformation of the sphere's equations from Cartesian to cylindrical. Use \( (r, \theta, z) \) for computation. 2. **Setting Up the Triple Integral:** - Construct the limits for \( r \), \( \theta \), and \( z \) based on the sphere's radius and center. - Define the density in terms of cylindrical coordinates. 3. **Integration:** - Integrate with respect to \( z \), \( r \), and \( \theta \) successively. - Compute the limits of integration from the boundaries of the sphere. 4. **Compute Mass:** - Evaluate the complete integral to find the total mass of the sphere. Understanding and applying these steps will enable you to calculate the mass using cylindrical coordinates effectively.
Transcribed Image Text:**Title:** Calculating the Mass of a Sphere Using Triple Integrals in Cylindrical Coordinates **Objective:** Learn how to use triple integrals in cylindrical coordinates to determine the mass of a given sphere. **Problem Statement:** Use a triple integral in cylindrical coordinates to find the mass of a sphere with a density function \( \delta = z \) and a radius of 2, centered at \( (2, 0, 0) \). **Solution Steps:** 1. **Convert to Cylindrical Coordinates:** - Consider the transformation of the sphere's equations from Cartesian to cylindrical. Use \( (r, \theta, z) \) for computation. 2. **Setting Up the Triple Integral:** - Construct the limits for \( r \), \( \theta \), and \( z \) based on the sphere's radius and center. - Define the density in terms of cylindrical coordinates. 3. **Integration:** - Integrate with respect to \( z \), \( r \), and \( \theta \) successively. - Compute the limits of integration from the boundaries of the sphere. 4. **Compute Mass:** - Evaluate the complete integral to find the total mass of the sphere. Understanding and applying these steps will enable you to calculate the mass using cylindrical coordinates effectively.
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Mass is calculated by tripple integral of density. 

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