Set up the iterated integral for evaluating f(r,0,z) dz r dr de over the given region D. D is the prism whose base is D the triangle in the xy-plane bounded by the x-axis and the lines y=x and x = 5 and whose top lies in the plane z = 6-y. 000 SSS f(r,0,z) dz r dr de C z=6-y - +3
Set up the iterated integral for evaluating f(r,0,z) dz r dr de over the given region D. D is the prism whose base is D the triangle in the xy-plane bounded by the x-axis and the lines y=x and x = 5 and whose top lies in the plane z = 6-y. 000 SSS f(r,0,z) dz r dr de C z=6-y - +3
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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![**Title: Setting Up Iterated Integrals for a Given Prism**
### Problem Statement:
Set up the iterated integral for evaluating
\[ \iiint\limits_D f(r, \theta, z) \, dz \, r \, dr \, d\theta \]
over the given region \( D \). \( D \) is the prism whose base is the triangle in the \( xy \)-plane bounded by the \( x \)-axis and the lines \( y = x \) and \( x = 5 \) and whose top lies in the plane \( z = 6 - y \).
### Steps to Set Up the Iterated Integral:
1. **Identify the Boundaries of the Region \( D \):**
- The base of the prism lies in the \( xy \)-plane and is bounded by:
- The \( x \)-axis (\( y = 0 \))
- The line \( y = x \)
- The vertical line \( x = 5 \)
- The top of the prism lies in the plane \( z = 6 - y \).
2. **Determine the Projection in the \( xy \)-Plane:**
The base of the region \( D \) is a triangular region with vertices at \( (0,0) \), \( (5,0) \), and \( (5,5) \). This triangular region can be described as:
\[
\begin{cases}
0 \le x \le 5 \\
0 \le y \le x
\end{cases}
\]
3. **Identify the Bounds for \( z \):**
Given the top plane \( z = 6 - y \), the bounds for \( z \) are from \( z = 0 \) (bottom in the \( xy \)-plane) to \( z = 6 - y \).
4. **Set Up the Iterated Integral:**
Combining these bounds, the iterated integral can be expressed as:
\[
\int_{0}^{5} \int_{0}^{x} \int_{0}^{6-y} f(r, \theta, z) \, dz \, dy \, dx
\]
To switch to cylindrical coordinates, we need to consider the radial component \( r \), and the angle \( \theta \). In this scenario](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1840ebe-b0df-4fe8-9210-d8e3dcfa32cc%2F6b4118f7-6baa-4dda-865f-24c2d0ed778f%2Fgsqiylk_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Setting Up Iterated Integrals for a Given Prism**
### Problem Statement:
Set up the iterated integral for evaluating
\[ \iiint\limits_D f(r, \theta, z) \, dz \, r \, dr \, d\theta \]
over the given region \( D \). \( D \) is the prism whose base is the triangle in the \( xy \)-plane bounded by the \( x \)-axis and the lines \( y = x \) and \( x = 5 \) and whose top lies in the plane \( z = 6 - y \).
### Steps to Set Up the Iterated Integral:
1. **Identify the Boundaries of the Region \( D \):**
- The base of the prism lies in the \( xy \)-plane and is bounded by:
- The \( x \)-axis (\( y = 0 \))
- The line \( y = x \)
- The vertical line \( x = 5 \)
- The top of the prism lies in the plane \( z = 6 - y \).
2. **Determine the Projection in the \( xy \)-Plane:**
The base of the region \( D \) is a triangular region with vertices at \( (0,0) \), \( (5,0) \), and \( (5,5) \). This triangular region can be described as:
\[
\begin{cases}
0 \le x \le 5 \\
0 \le y \le x
\end{cases}
\]
3. **Identify the Bounds for \( z \):**
Given the top plane \( z = 6 - y \), the bounds for \( z \) are from \( z = 0 \) (bottom in the \( xy \)-plane) to \( z = 6 - y \).
4. **Set Up the Iterated Integral:**
Combining these bounds, the iterated integral can be expressed as:
\[
\int_{0}^{5} \int_{0}^{x} \int_{0}^{6-y} f(r, \theta, z) \, dz \, dy \, dx
\]
To switch to cylindrical coordinates, we need to consider the radial component \( r \), and the angle \( \theta \). In this scenario
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