Evaluate the triple integral of f(x, y, z) = sin(x² + y²) over the solid cylinder with height 7 and with base of radius 2 centered on the z axis at z = -1. Integral =|
Evaluate the triple integral of f(x, y, z) = sin(x² + y²) over the solid cylinder with height 7 and with base of radius 2 centered on the z axis at z = -1. Integral =|
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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![**Triple Integral Evaluation over a Cylindrical Region**
**Problem Statement:**
Evaluate the triple integral of \( f(x, y, z) = \sin(x^2 + y^2) \) over the solid cylinder with height 7 and base of radius 2 centered on the \(z\)-axis at \( z = -1 \).
**Mathematical Representation:**
\[
\text{Integral} = \int \int \int_V f(x, y, z) \, dV
\]
Where:
- \(V\) represents the volume of the cylinder,
- \(f(x, y, z) = \sin(x^2 + y^2)\).
*Note:* The volume \(V\) is defined by the cylindrical coordinates, where the base of the cylinder has a radius of 2 and is centered on the \(z\)-axis at \( z = -1 \), with a height of 7.
To input your solution, please use the provided field labeled "Integral = " to type your calculated result or derivation.
---
*Explanation:*
This problem involves evaluating a triple integral over a cylindrical volume. You will need to set up the integral in cylindrical coordinates for easier computation. Here, the height of the cylinder implies that \(z\) ranges from \(-1\) to \(6\) (as it starts from \(z = -1\) and extends 7 units upward). The radius \(r\) ranges from \(0\) to \(2\), and \(\theta\) ranges from \(0\) to \(2\pi\).
The integral can be expressed as:
\[
\int_{z=-1}^{6} \int_{r=0}^{2} \int_{\theta=0}^{2\pi} \sin(r^2) \, r \, d\theta \, dr \, dz
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Transcribed Image Text:**Triple Integral Evaluation over a Cylindrical Region**
**Problem Statement:**
Evaluate the triple integral of \( f(x, y, z) = \sin(x^2 + y^2) \) over the solid cylinder with height 7 and base of radius 2 centered on the \(z\)-axis at \( z = -1 \).
**Mathematical Representation:**
\[
\text{Integral} = \int \int \int_V f(x, y, z) \, dV
\]
Where:
- \(V\) represents the volume of the cylinder,
- \(f(x, y, z) = \sin(x^2 + y^2)\).
*Note:* The volume \(V\) is defined by the cylindrical coordinates, where the base of the cylinder has a radius of 2 and is centered on the \(z\)-axis at \( z = -1 \), with a height of 7.
To input your solution, please use the provided field labeled "Integral = " to type your calculated result or derivation.
---
*Explanation:*
This problem involves evaluating a triple integral over a cylindrical volume. You will need to set up the integral in cylindrical coordinates for easier computation. Here, the height of the cylinder implies that \(z\) ranges from \(-1\) to \(6\) (as it starts from \(z = -1\) and extends 7 units upward). The radius \(r\) ranges from \(0\) to \(2\), and \(\theta\) ranges from \(0\) to \(2\pi\).
The integral can be expressed as:
\[
\int_{z=-1}^{6} \int_{r=0}^{2} \int_{\theta=0}^{2\pi} \sin(r^2) \, r \, d\theta \, dr \, dz
\]
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