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 \(\iiint 6 \, dA\), where \(\mathcal{R}\) is the region given by:
\[ \{(x,y,z) : 1 \leq x^2 + y^2 \leq 16, x^2 + y^2 \leq z \leq 16 - x^2 - y^2 \} \]
#### Explanation of Region \(\mathcal{R}\):
The region \(\mathcal{R}\) is defined by a combination of inequalities involving \(x\), \(y\), and \(z\):
1. \(1 \leq x^2 + y^2 \leq 16\):
This inequality describes the region in the \(xy\)-plane bounded between two circles. The inner circle has a radius of 1 and the outer circle has a radius of 4 (since \(4^2 = 16\)).
2. \(x^2 + y^2 \leq z \leq 16 - x^2 - y^2\):
This describes the bounds for \(z\) in terms of the values of \(x^2 + y^2\):
- The lower bound for \(z\) is the value of \(x^2 + y^2\).
- The upper bound for \(z\) is \(16 - x^2 + y^2\).
This region forms a solid bounded by these surfaces in the space.
There are no graphs or diagrams included in the image. If this were implemented on a website, it might be helpful to illustrate the region with a 3D graph showing the boundaries described above.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9d15e77-8c44-4e48-af9a-bae33a9e346c%2Fdf4dbec7-76a8-4824-b31d-4e3cfe66a9dd%2F6ouyddr_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Evaluate the triple integral \(\iiint 6 \, dA\), where \(\mathcal{R}\) is the region given by:
\[ \{(x,y,z) : 1 \leq x^2 + y^2 \leq 16, x^2 + y^2 \leq z \leq 16 - x^2 - y^2 \} \]
#### Explanation of Region \(\mathcal{R}\):
The region \(\mathcal{R}\) is defined by a combination of inequalities involving \(x\), \(y\), and \(z\):
1. \(1 \leq x^2 + y^2 \leq 16\):
This inequality describes the region in the \(xy\)-plane bounded between two circles. The inner circle has a radius of 1 and the outer circle has a radius of 4 (since \(4^2 = 16\)).
2. \(x^2 + y^2 \leq z \leq 16 - x^2 - y^2\):
This describes the bounds for \(z\) in terms of the values of \(x^2 + y^2\):
- The lower bound for \(z\) is the value of \(x^2 + y^2\).
- The upper bound for \(z\) is \(16 - x^2 + y^2\).
This region forms a solid bounded by these surfaces in the space.
There are no graphs or diagrams included in the image. If this were implemented on a website, it might be helpful to illustrate the region with a 3D graph showing the boundaries described above.
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