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Use cylindrical coordinates and a triple integral to find the mass of a solid bounded by the xy-plane and the paraboloid z = 4 – r² whose density is f(r,8,z) = 5 – z (denser near the base).

Use cylindrical coordinates and a triple integral to find the mass of a solid bounded by the xy-plane and the paraboloid z = 4 – r² whose density is f(r,8,z) = 5 – z (denser near the base).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Title: Calculating the Mass of a Solid Using Cylindrical Coordinates** **Introduction:** In this exercise, we will learn how to calculate the mass of a solid bounded by the xy-plane and a paraboloid using cylindrical coordinates and a triple integral. The paraboloid is defined by the equation \( z = 4 - r^2 \). The density of the solid is given by \( f(r, \theta, z) = 5 - z \), indicating that the density increases as we approach the base of the solid. **Problem Statement:** *Calculate the mass of the solid bounded by the following conditions:* - **Boundary**: The solid is bounded below by the xy-plane and above by the paraboloid \( z = 4 - r^2 \). - **Density Function**: The density at any point within the solid is given by \( f(r, \theta, z) = 5 - z \). **Approach:** We will use cylindrical coordinates \((r, \theta, z)\) and set up a triple integral to find the mass. The integration will cover: 1. The radial component \( r \), which extends from 0 to the maximum radius of the paraboloid boundary. 2. The angular component \( \theta \), which spans from 0 to \( 2\pi \) for full rotation around the z-axis. 3. The vertical component \( z \), which is bounded by the xy-plane and the paraboloid surface \( z = 4 - r^2 \). **Conclusion:** This method allows us to determine the mass by evaluating the triple integral \( \int \int \int f(r, \theta, z) \, dz \, dr \, d\theta \), taking into account the varying density throughout the solid. Through this approach, we can appreciate the utility of cylindrical coordinates in simplifying the integration of solids with rotational symmetry.
Transcribed Image Text:**Title: Calculating the Mass of a Solid Using Cylindrical Coordinates** **Introduction:** In this exercise, we will learn how to calculate the mass of a solid bounded by the xy-plane and a paraboloid using cylindrical coordinates and a triple integral. The paraboloid is defined by the equation \( z = 4 - r^2 \). The density of the solid is given by \( f(r, \theta, z) = 5 - z \), indicating that the density increases as we approach the base of the solid. **Problem Statement:** *Calculate the mass of the solid bounded by the following conditions:* - **Boundary**: The solid is bounded below by the xy-plane and above by the paraboloid \( z = 4 - r^2 \). - **Density Function**: The density at any point within the solid is given by \( f(r, \theta, z) = 5 - z \). **Approach:** We will use cylindrical coordinates \((r, \theta, z)\) and set up a triple integral to find the mass. The integration will cover: 1. The radial component \( r \), which extends from 0 to the maximum radius of the paraboloid boundary. 2. The angular component \( \theta \), which spans from 0 to \( 2\pi \) for full rotation around the z-axis. 3. The vertical component \( z \), which is bounded by the xy-plane and the paraboloid surface \( z = 4 - r^2 \). **Conclusion:** This method allows us to determine the mass by evaluating the triple integral \( \int \int \int f(r, \theta, z) \, dz \, dr \, d\theta \), taking into account the varying density throughout the solid. Through this approach, we can appreciate the utility of cylindrical coordinates in simplifying the integration of solids with rotational symmetry.
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