Find the volume of the solid generated by revolving the region about the given line.a.) State the volume method usedb.) Give exact answer, please include sketch of the region **Mathematical Region Bounded by Curves**The area of interest is defined by the curves \( y = 7x - x^2 \) and \( y = x \), and rotates about the vertical line \( x = 6 \). Using integration techniques, we can find various geometrical properties such as the area, volume, or centroid of this region.- \( y = 7x - x^2 \): This is a quadratic equation representing a downward-opening parabola.- \( y = x \): This is a linear equation representing a straight line with a positive slope.To address this problem, consider the following steps:1. **Find the Intersection Points**: Determine where the two functions intersect by setting \( 7x - x^2 = x \) and solving for \( x \).2. **Set Up the Integral**: Depending on the property (area, volume, etc.), set up the appropriate integral.3. **Integrate**: Perform the integration process and solve the integral analytically.4. **Rotation About the Line \( x = 6 \)**: If calculating the volume of revolution, apply the method of disks or shells to account for rotation around \( x = 6 \).By understanding and following these steps, one can solve for the desired geometrical property for the given region. This exercise integrates knowledge from calculus and geometry to solve real-world problems.

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The region bounded by y = 7x - x² and y = x about the line x = 6.

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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Find the volume of the solid generated by revolving the region about the given line. a.) State the volume method used b.) Give exact answer, please include sketch of the region

**Mathematical Region Bounded by Curves**

The area of interest is defined by the curves \( y = 7x - x^2 \) and \( y = x \), and rotates about the vertical line \( x = 6 \). Using integration techniques, we can find various geometrical properties such as the area, volume, or centroid of this region.

- \( y = 7x - x^2 \): This is a quadratic equation representing a downward-opening parabola.
- \( y = x \): This is a linear equation representing a straight line with a positive slope.

To address this problem, consider the following steps:
1. **Find the Intersection Points**: Determine where the two functions intersect by setting \( 7x - x^2 = x \) and solving for \( x \).
2. **Set Up the Integral**: Depending on the property (area, volume, etc.), set up the appropriate integral.
3. **Integrate**: Perform the integration process and solve the integral analytically.
4. **Rotation About the Line \( x = 6 \)**: If calculating the volume of revolution, apply the method of disks or shells to account for rotation around \( x = 6 \).

By understanding and following these steps, one can solve for the desired geometrical property for the given region. This exercise integrates knowledge from calculus and geometry to solve real-world problems.

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