A region bounded by f(x) 8 7 6 Find the volume of the solid formed by revolving the region about the y-axis. 5 4 3 2 1 X -2-11 1 2 3 4 5 6 7 8 -2 O V = 64 V = V = 364 3 193 3 OV=81 TU = T -1/2 x x + 5, y = 1, y = 4, and x = 1 is shown below.
A region bounded by f(x) 8 7 6 Find the volume of the solid formed by revolving the region about the y-axis. 5 4 3 2 1 X -2-11 1 2 3 4 5 6 7 8 -2 O V = 64 V = V = 364 3 193 3 OV=81 TU = T -1/2 x x + 5, y = 1, y = 4, and x = 1 is shown below.
Calculus: Early Transcendentals
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Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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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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![### Calculating the Volume of a Solid of Revolution
**Problem Statement:**
A region bounded by \( f(x) = -\frac{1}{2} x + 5 \), \( y = 1 \), \( y = 4 \), and \( x = 1 \) is shown below.
**Objective:**
Find the volume of the solid formed by revolving the region about the y-axis.
**Graph Explanation:**
The provided graph features:
- Two axes: X (horizontal) and Y (vertical), labeled accordingly.
- The boundaries of the region of interest:
- **Function**: \( f(x) = -\frac{1}{2} x + 5 \), which is a straight line with a negative slope intersecting the Y-axis at \( y = 5 \) and decreasing to the right.
- **Horizontal lines**: \( y = 1 \) and \( y = 4 \), marking the lower and upper limits of the region.
- **Vertical line**: \( x = 1 \), marking the right boundary of the region.
The shaded region of interest is enclosed by these boundaries and represents the area that will be revolved around the y-axis to form a solid.
**Volume Calculation Options:**
You need to determine which among the following choices correctly represents the volume (V) of the solid:
- V = \( 64\pi \)
- V = \( \frac{364\pi}{3} \)
- V = \( \frac{193\pi}{3} \)
- V = \( 81\pi \)
To solve this problem, you would typically integrate the given function between the appropriate bounds, considering the method of cylindrical shells or another suitable method for volumes of revolution.
**Solution Approach:**
1. **Identify bounds** for integration:
- From \( y = 1 \) to \( y = 4 \).
2. **Express \( x \) as a function of \( y \)**:
- Given \( f(x) = -\frac{1}{2}x + 5 \), solve for \( x \): \( x = 2(5 - y) \).
3. **Set up the integral** for volume using the cylindrical shell method:
\[
V = 2\pi \int_{1}^{4} (radius)(height) \, dy
\]
\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6539f692-5656-46fd-b828-f0fe9bb331a2%2F08593063-8a79-4639-9ceb-3b90aa4a673a%2Fqo9y0x_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating the Volume of a Solid of Revolution
**Problem Statement:**
A region bounded by \( f(x) = -\frac{1}{2} x + 5 \), \( y = 1 \), \( y = 4 \), and \( x = 1 \) is shown below.
**Objective:**
Find the volume of the solid formed by revolving the region about the y-axis.
**Graph Explanation:**
The provided graph features:
- Two axes: X (horizontal) and Y (vertical), labeled accordingly.
- The boundaries of the region of interest:
- **Function**: \( f(x) = -\frac{1}{2} x + 5 \), which is a straight line with a negative slope intersecting the Y-axis at \( y = 5 \) and decreasing to the right.
- **Horizontal lines**: \( y = 1 \) and \( y = 4 \), marking the lower and upper limits of the region.
- **Vertical line**: \( x = 1 \), marking the right boundary of the region.
The shaded region of interest is enclosed by these boundaries and represents the area that will be revolved around the y-axis to form a solid.
**Volume Calculation Options:**
You need to determine which among the following choices correctly represents the volume (V) of the solid:
- V = \( 64\pi \)
- V = \( \frac{364\pi}{3} \)
- V = \( \frac{193\pi}{3} \)
- V = \( 81\pi \)
To solve this problem, you would typically integrate the given function between the appropriate bounds, considering the method of cylindrical shells or another suitable method for volumes of revolution.
**Solution Approach:**
1. **Identify bounds** for integration:
- From \( y = 1 \) to \( y = 4 \).
2. **Express \( x \) as a function of \( y \)**:
- Given \( f(x) = -\frac{1}{2}x + 5 \), solve for \( x \): \( x = 2(5 - y) \).
3. **Set up the integral** for volume using the cylindrical shell method:
\[
V = 2\pi \int_{1}^{4} (radius)(height) \, dy
\]
\
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