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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![### Problem Statement
Evaluate the triple integral:
\[ \iiint_E \frac{1}{x^2 + y^2 + z^2} \, dV \]
where \( E \) lies between the spheres \( x^2 + y^2 + z^2 = 4 \) and \( x^2 + y^2 + z^2 = 16 \) in the first octant.
### Explanation
This problem involves evaluating a triple integral within a specific region \( E \). The region \( E \) is defined as the space between two concentric spheres with radii 2 and 4 (since \( \sqrt{4} = 2 \) and \( \sqrt{16} = 4 \)), restricted to the first octant.
In spherical coordinates, a point in 3D space is represented as:
- \( \rho \): the radial distance from the origin,
- \( \theta \): the azimuthal angle in the \( xy \)-plane from the positive \( x \)-axis, and
- \( \phi \): the polar angle from the positive \( z \)-axis.
For the given problem, the limits for \( \rho \), \( \theta \), and \( \phi \) can be outlined as follows:
- \( \rho \) ranges from 2 to 4 (since it is constrained between the spheres),
- \( \theta \) ranges from 0 to \(\frac{\pi}{2} \) (first octant in the \( xy \)-plane),
- \( \phi \) ranges from 0 to \(\frac{\pi}{2} \) (first octant above the \( xy \)-plane).
The integrand in spherical coordinates \( \frac{1}{x^2 + y^2 + z^2} \) becomes \( \frac{1}{\rho^2} \).
Hence, the integral becomes:
\[ \iiint_E \frac{1}{\rho^2} \, \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta \]
Simplifying this:
\[ \iiint_E \sin \phi \, d\rho \, d\phi \, d\theta \]
### Solution Outline
1. **Convert Cartesian coordinates to spherical coordinates**.
2. **Set up the integral with the new bounds and transformations](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F92ef1733-4d13-4c5f-b698-2777de49beac%2Ff802303f-e52a-4d02-ba19-25426fa989ff%2Faxxulzk_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Evaluate the triple integral:
\[ \iiint_E \frac{1}{x^2 + y^2 + z^2} \, dV \]
where \( E \) lies between the spheres \( x^2 + y^2 + z^2 = 4 \) and \( x^2 + y^2 + z^2 = 16 \) in the first octant.
### Explanation
This problem involves evaluating a triple integral within a specific region \( E \). The region \( E \) is defined as the space between two concentric spheres with radii 2 and 4 (since \( \sqrt{4} = 2 \) and \( \sqrt{16} = 4 \)), restricted to the first octant.
In spherical coordinates, a point in 3D space is represented as:
- \( \rho \): the radial distance from the origin,
- \( \theta \): the azimuthal angle in the \( xy \)-plane from the positive \( x \)-axis, and
- \( \phi \): the polar angle from the positive \( z \)-axis.
For the given problem, the limits for \( \rho \), \( \theta \), and \( \phi \) can be outlined as follows:
- \( \rho \) ranges from 2 to 4 (since it is constrained between the spheres),
- \( \theta \) ranges from 0 to \(\frac{\pi}{2} \) (first octant in the \( xy \)-plane),
- \( \phi \) ranges from 0 to \(\frac{\pi}{2} \) (first octant above the \( xy \)-plane).
The integrand in spherical coordinates \( \frac{1}{x^2 + y^2 + z^2} \) becomes \( \frac{1}{\rho^2} \).
Hence, the integral becomes:
\[ \iiint_E \frac{1}{\rho^2} \, \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta \]
Simplifying this:
\[ \iiint_E \sin \phi \, d\rho \, d\phi \, d\theta \]
### Solution Outline
1. **Convert Cartesian coordinates to spherical coordinates**.
2. **Set up the integral with the new bounds and transformations
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