Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis. y = 4(3 – x) y = 0 X = 0
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis. y = 4(3 – x) y = 0 X = 0
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
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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![**Problem Statement:**
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis.
**Equations:**
\[ y = 4(3 - x) \]
\[ y = 0 \]
\[ x = 0 \]
**Explanation:**
To solve this problem, you calculate the volume of a solid of revolution by using the method of disks or washers. In this specific instance, since the region is rotated about the y-axis, you generally use the method of cylindrical shells. Here’s a detailed explanation of the components involved:
1. **Equation of the Line:**
- \( y = 4(3 - x) \) is a linear equation representing a line when plotted on a graph.
- This line has a slope of \(-4\) and a y-intercept at \( y = 12 \).
2. **Boundaries:**
- \( y = 0 \) is the x-axis, serving as a lower boundary for the region.
- \( x = 0 \) is the y-axis, serving as one side boundary for the region.
3. **Region of Interest:**
- The bounded region is a triangular area under the line \( y = 4(3-x) \), above the x-axis, and to the right of the y-axis.
4. **Volume Calculation:**
- Apply the method of cylindrical shells to find the volume of the solid formed by revolving this region around the y-axis.
- The volume \( V \) can be calculated using the integral:
\[
V = \int_{a}^{b} 2\pi \cdot x \cdot f(x) \, dx
\]
In this case, \( f(x) = 4(3 - x) \), and you need to determine the limits of integration from the intersection points of the given equations.
This problem challenges your understanding of integration and geometric transformations, crucial for applications in calculus and physics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25721e20-facb-4d0d-8f2d-fb325bdcbfad%2F9feb2001-c0fb-42db-a020-2e812e10d9e2%2F2z2350f_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis.
**Equations:**
\[ y = 4(3 - x) \]
\[ y = 0 \]
\[ x = 0 \]
**Explanation:**
To solve this problem, you calculate the volume of a solid of revolution by using the method of disks or washers. In this specific instance, since the region is rotated about the y-axis, you generally use the method of cylindrical shells. Here’s a detailed explanation of the components involved:
1. **Equation of the Line:**
- \( y = 4(3 - x) \) is a linear equation representing a line when plotted on a graph.
- This line has a slope of \(-4\) and a y-intercept at \( y = 12 \).
2. **Boundaries:**
- \( y = 0 \) is the x-axis, serving as a lower boundary for the region.
- \( x = 0 \) is the y-axis, serving as one side boundary for the region.
3. **Region of Interest:**
- The bounded region is a triangular area under the line \( y = 4(3-x) \), above the x-axis, and to the right of the y-axis.
4. **Volume Calculation:**
- Apply the method of cylindrical shells to find the volume of the solid formed by revolving this region around the y-axis.
- The volume \( V \) can be calculated using the integral:
\[
V = \int_{a}^{b} 2\pi \cdot x \cdot f(x) \, dx
\]
In this case, \( f(x) = 4(3 - x) \), and you need to determine the limits of integration from the intersection points of the given equations.
This problem challenges your understanding of integration and geometric transformations, crucial for applications in calculus and physics.
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