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6dA, where R is the region given by { (x,y,z): 1sx²+y?s16, x²+y²szs16-x²-y2}. Evaluate

6dA, where R is the region given by { (x,y,z): 1sx²+y?s16, x²+y²szs16-x²-y2}. Evaluate

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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### 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.
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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