Use the method of cylindrical shells to find the volume generated by rotating the region bounded by y = 10x - x², y = x about the line x = 12.
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by y = 10x - x², y = x about the line x = 12.
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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![**Problem Statement: Application of Cylindrical Shells Method**
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves \( y = 10x - x^2 \) and \( y = x \) about the line \( x = 12 \).
**Solution Steps:**
1. **Identify the Region:**
- The region of interest is bounded by the two functions \( y = 10x - x^2 \) and \( y = x \).
- Solve for the points of intersection by equating the two equations:
\[ 10x - x^2 = x \]
\[ x^2 - 9x = 0 \]
\[ x(x - 9) = 0 \]
- Thus, the points of intersection are \( x = 0 \) and \( x = 9 \).
2. **Cylindrical Shells Method:**
- We rotate about the line \( x = 12 \).
- The radius \( r \) of a shell at a point \( x \) is the distance from the line \( x = 12 \) to \( x \), which is \( r = 12 - x \).
- The height \( h \) of each shell is the difference in \( y \)-values: \( h = (10x - x^2) - x \).
- Therefore, the height is \( h = 9x - x^2 \).
3. **Integrate to Find the Volume:**
- The formula for the volume of a cylindrical shell is:
\[ V = 2\pi \int_{a}^{b} (radius)(height) \, dx \]
- Here it becomes:
\[ V = 2\pi \int_{0}^{9} (12 - x)(9x - x^2) \, dx \]
- Evaluate this integral to find the volume.
By setting up this integral, you can compute the exact volume of the solid generated by the rotation about the specified line. If further calculation is needed, you can expand and integrate the polynomial.
This exploration demonstrates the practical application of the cylindrical shells method in calculus, particularly in problems involving solid revolutions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F853bd72d-1b9a-40ff-a061-0c00f168c5bc%2F29e53372-81a2-462c-a02a-0dde9104ccbd%2Fwwr7yk7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Application of Cylindrical Shells Method**
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves \( y = 10x - x^2 \) and \( y = x \) about the line \( x = 12 \).
**Solution Steps:**
1. **Identify the Region:**
- The region of interest is bounded by the two functions \( y = 10x - x^2 \) and \( y = x \).
- Solve for the points of intersection by equating the two equations:
\[ 10x - x^2 = x \]
\[ x^2 - 9x = 0 \]
\[ x(x - 9) = 0 \]
- Thus, the points of intersection are \( x = 0 \) and \( x = 9 \).
2. **Cylindrical Shells Method:**
- We rotate about the line \( x = 12 \).
- The radius \( r \) of a shell at a point \( x \) is the distance from the line \( x = 12 \) to \( x \), which is \( r = 12 - x \).
- The height \( h \) of each shell is the difference in \( y \)-values: \( h = (10x - x^2) - x \).
- Therefore, the height is \( h = 9x - x^2 \).
3. **Integrate to Find the Volume:**
- The formula for the volume of a cylindrical shell is:
\[ V = 2\pi \int_{a}^{b} (radius)(height) \, dx \]
- Here it becomes:
\[ V = 2\pi \int_{0}^{9} (12 - x)(9x - x^2) \, dx \]
- Evaluate this integral to find the volume.
By setting up this integral, you can compute the exact volume of the solid generated by the rotation about the specified line. If further calculation is needed, you can expand and integrate the polynomial.
This exploration demonstrates the practical application of the cylindrical shells method in calculus, particularly in problems involving solid revolutions.
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