Use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the x-axis. y = 7 - x x 1 --2 1 2 3 4 5 6 7 8 9 10 11
Use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the x-axis. y = 7 - x x 1 --2 1 2 3 4 5 6 7 8 9 10 11
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Using the Shell Method to Find Volume**
To find the volume of a solid generated by revolving a plane region about the x-axis, one can use the shell method. Consider the line equation:
\[ y = 7 - x \]
**Graph Explanation:**
- The graph shows a region bounded by the line \( y = 7 - x \), the x-axis, and vertical lines at \( x = 7 \) and \( x = 10 \).
- The region is shaded blue, indicating the area to be revolved.
- A cylindrical shell is highlighted in yellow, illustrating the shell method.
**Shell Method Concept:**
The shell method involves using cylindrical shells to approximate the volume of the solid. The general formula for the volume using the shell method is:
\[ V = 2\pi \int_{a}^{b} (radius)(height) \, dx \]
In this context:
- **Radius:** The distance from the shell to the axis of rotation, which is the y-coordinate of the function.
- **Height:** Evaluated as the value of the function \( y = 7 - x \).
**Steps:**
1. Determine the limits of integration, which are the x-values where the region is bounded.
2. Set up the integral using the shell method formula.
3. Evaluate the integral to find the volume of the solid.
For further help and resources, click on "Need Help? Read It!" links provided on the webpage.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe35298e5-10ed-452c-b18c-35d9ef29ce25%2F8a6b0458-65b6-480d-87b6-1041e3bdcd9d%2F2s18mqf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Using the Shell Method to Find Volume**
To find the volume of a solid generated by revolving a plane region about the x-axis, one can use the shell method. Consider the line equation:
\[ y = 7 - x \]
**Graph Explanation:**
- The graph shows a region bounded by the line \( y = 7 - x \), the x-axis, and vertical lines at \( x = 7 \) and \( x = 10 \).
- The region is shaded blue, indicating the area to be revolved.
- A cylindrical shell is highlighted in yellow, illustrating the shell method.
**Shell Method Concept:**
The shell method involves using cylindrical shells to approximate the volume of the solid. The general formula for the volume using the shell method is:
\[ V = 2\pi \int_{a}^{b} (radius)(height) \, dx \]
In this context:
- **Radius:** The distance from the shell to the axis of rotation, which is the y-coordinate of the function.
- **Height:** Evaluated as the value of the function \( y = 7 - x \).
**Steps:**
1. Determine the limits of integration, which are the x-values where the region is bounded.
2. Set up the integral using the shell method formula.
3. Evaluate the integral to find the volume of the solid.
For further help and resources, click on "Need Help? Read It!" links provided on the webpage.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
