Evaluate the triple integral = 2. B = {(x, y, z) | x² + y² ≤ 7², x ≥ 0, y ≥ 0, 0 ≤ z ≤ 10} and g(x, y, z) Z 1.0 0.75 0.5 0.25 0.0 g(x, y, z) dV over solid B. B 0 1 y 2 O Note: The graph is an example. The scale may not be the same for your particular problem. Give your answer accurate to at least three decimal places. Hint: Convert from rectangular to cylindrical coordinate system.
Evaluate the triple integral = 2. B = {(x, y, z) | x² + y² ≤ 7², x ≥ 0, y ≥ 0, 0 ≤ z ≤ 10} and g(x, y, z) Z 1.0 0.75 0.5 0.25 0.0 g(x, y, z) dV over solid B. B 0 1 y 2 O Note: The graph is an example. The scale may not be the same for your particular problem. Give your answer accurate to at least three decimal places. Hint: Convert from rectangular to cylindrical coordinate system.
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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![**Evaluate the Triple Integral**
Evaluate the triple integral \(\iiint_B g(x, y, z) \, dV\) over solid \(B\).
\[ B = \{(x, y, z) \mid x^2 + y^2 \leq r^2, x \geq 0, 0 \leq z \leq 10\} \text{ and } g(x, y, z) = z \]
The graph represents a three-dimensional region defined by the inequalities above. The solid \(B\) is bounded by the cylinder \(x^2 + y^2 \leq r^2\) for \(x \geq 0\), stretching from \(z = 0\) to \(z = 10\). The function \(g(x, y, z) = z\) is integrated over this volume.
**Graph Explanation**
- **Axes**: The \(x\), \(y\), and \(z\) axes are labeled, with \(z\) ranging from 0 to 1, \(x\) from 0 to approximately 3, and \(y\) from 0 to 1.
- **Shape**: The shape represents a quarter-cylinder extending along the positive \(x\) direction.
- **Surface**: The top surface of the quarter-cylinder is flat, consistent with the function \(g(x, y, z) = z\).
**Note**: The graph is an example. The scale may not be the same for your particular problem.
Give your answer accurate to at least three decimal places.
**Hint**: Convert from rectangular to cylindrical coordinate system.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8216494-bf6c-44b6-949f-dd8ff9f3594a%2Ff264fbcd-eecb-4a24-9e4a-50443ea25c75%2Fpjo4erf_processed.png&w=3840&q=75)
Transcribed Image Text:**Evaluate the Triple Integral**
Evaluate the triple integral \(\iiint_B g(x, y, z) \, dV\) over solid \(B\).
\[ B = \{(x, y, z) \mid x^2 + y^2 \leq r^2, x \geq 0, 0 \leq z \leq 10\} \text{ and } g(x, y, z) = z \]
The graph represents a three-dimensional region defined by the inequalities above. The solid \(B\) is bounded by the cylinder \(x^2 + y^2 \leq r^2\) for \(x \geq 0\), stretching from \(z = 0\) to \(z = 10\). The function \(g(x, y, z) = z\) is integrated over this volume.
**Graph Explanation**
- **Axes**: The \(x\), \(y\), and \(z\) axes are labeled, with \(z\) ranging from 0 to 1, \(x\) from 0 to approximately 3, and \(y\) from 0 to 1.
- **Shape**: The shape represents a quarter-cylinder extending along the positive \(x\) direction.
- **Surface**: The top surface of the quarter-cylinder is flat, consistent with the function \(g(x, y, z) = z\).
**Note**: The graph is an example. The scale may not be the same for your particular problem.
Give your answer accurate to at least three decimal places.
**Hint**: Convert from rectangular to cylindrical coordinate system.
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