Use polar coordinates to find the volume of the given solid. Below the plane 6x + y + z = 8 and above the disk x2 + y2 s 1
Use polar coordinates to find the volume of the given solid. Below the plane 6x + y + z = 8 and above the disk x2 + y2 s 1
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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![### Finding the Volume of the Given Solid Using Polar Coordinates
To find the volume of the solid bounded below the plane \(6x + y + z = 8\) and above the disk \(x^2 + y^2 \leq 1\), we can use polar coordinates for simplification.
### Problem Statement
We want to determine the volume of the solid that lies:
- Below the plane: \(6x + y + z = 8\)
- Above the disk: \(x^2 + y^2 \leq 1\)
### Key Steps:
1. **Convert Cartesian Coordinates to Polar Coordinates:**
- \(x = r \cos \theta\)
- \(y = r \sin \theta\)
- \(r^2 = x^2 + y^2\)
2. **Express the Equation of the Plane in Polar Coordinates:**
Given the plane equation \(6x + y + z = 8\), substitute \(x\) and \(y\):
\[
6(r \cos \theta) + r \sin \theta + z = 8
\]
\[
z = 8 - 6r \cos \theta - r \sin \theta
\]
3. **Set Up the Integral:**
- The region in the \(xy\)-plane is a disk with radius 1.
- The limits for \(r\) are from 0 to 1.
- The limits for \(\theta\) are from 0 to \(2\pi\).
The volume integral in polar coordinates becomes:
\[
V = \int_{0}^{2\pi} \int_{0}^{1} \left(8 - 6r \cos \theta - r \sin \theta\right) r \, dr \, d\theta
\]
4. **Evaluate the Integral:**
- Integrate with respect to \(r\) first.
- Then integrate with respect to \(\theta\).
### Graphical Representation
- There is a small rectangular area with a red cross below it, which might be an indicator for students to notice a detail or correction in a given context.
By analyzing the components and evaluating the integral correctly, you will find the desired volume of the solid.
<figure>
<img src="URL" alt="](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6615292-9c23-445f-b631-819ab1904cf2%2F9001cc4d-07be-44de-93c1-3cc4ecb7e80f%2Fxfrb66.png&w=3840&q=75)
Transcribed Image Text:### Finding the Volume of the Given Solid Using Polar Coordinates
To find the volume of the solid bounded below the plane \(6x + y + z = 8\) and above the disk \(x^2 + y^2 \leq 1\), we can use polar coordinates for simplification.
### Problem Statement
We want to determine the volume of the solid that lies:
- Below the plane: \(6x + y + z = 8\)
- Above the disk: \(x^2 + y^2 \leq 1\)
### Key Steps:
1. **Convert Cartesian Coordinates to Polar Coordinates:**
- \(x = r \cos \theta\)
- \(y = r \sin \theta\)
- \(r^2 = x^2 + y^2\)
2. **Express the Equation of the Plane in Polar Coordinates:**
Given the plane equation \(6x + y + z = 8\), substitute \(x\) and \(y\):
\[
6(r \cos \theta) + r \sin \theta + z = 8
\]
\[
z = 8 - 6r \cos \theta - r \sin \theta
\]
3. **Set Up the Integral:**
- The region in the \(xy\)-plane is a disk with radius 1.
- The limits for \(r\) are from 0 to 1.
- The limits for \(\theta\) are from 0 to \(2\pi\).
The volume integral in polar coordinates becomes:
\[
V = \int_{0}^{2\pi} \int_{0}^{1} \left(8 - 6r \cos \theta - r \sin \theta\right) r \, dr \, d\theta
\]
4. **Evaluate the Integral:**
- Integrate with respect to \(r\) first.
- Then integrate with respect to \(\theta\).
### Graphical Representation
- There is a small rectangular area with a red cross below it, which might be an indicator for students to notice a detail or correction in a given context.
By analyzing the components and evaluating the integral correctly, you will find the desired volume of the solid.
<figure>
<img src="URL" alt="
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