Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the y-axis. y = ₁, y = 0, x₁ = ¹, X₂ = 6 V=
Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the y-axis. y = ₁, y = 0, x₁ = ¹, X₂ = 6 V=
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
![**Problem Statement:**
Use the method of cylindrical shells to find the volume \( V \) generated by rotating the region bounded by the given curves about the y-axis.
Given:
\[ y = \frac{7}{x}, \quad y = 0, \quad x_1 = 1, \quad x_2 = 6 \]
**Calculation:**
\[ V = \]
(_Provide space or a formula input box for students to input their answer._)
Explanation:
To find the volume using the method of cylindrical shells, integrate the product of the circumference of a shell, the height of the shell, and the thickness of the shell along the x-axis. The cylindrical shell method uses the following formula:
\[ V = 2\pi \int_{a}^{b} (radius \cdot height \cdot thickness) \, dx \]
Where:
- The radius of the shell is the distance from the axis of rotation (y-axis), which is \( x \).
- The height of the shell is the function \( y = \frac{7}{x} \).
- The thickness is \( dx \).
Thus, the volume becomes:
\[ V = 2\pi \int_{1}^{6} x \left(\frac{7}{x}\right) dx \]
Simplifying the integrand:
\[ V = 2\pi \int_{1}^{6} 7 \, dx \]
\[ V = 2\pi \left[ 7x \right]_{1}^{6} \]
Evaluating the definite integral:
\[ V = 2\pi \left[ 7(6) - 7(1) \right] \]
\[ V = 2\pi \left[ 42 - 7 \right] \]
\[ V = 2\pi (35) \]
\[ V = 70\pi \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2e97f32-5988-4deb-9547-eb6ce37eb1f3%2Ff9bad6f4-026d-4fcd-a39b-516fe4ce4d6e%2Far9gdbu_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use the method of cylindrical shells to find the volume \( V \) generated by rotating the region bounded by the given curves about the y-axis.
Given:
\[ y = \frac{7}{x}, \quad y = 0, \quad x_1 = 1, \quad x_2 = 6 \]
**Calculation:**
\[ V = \]
(_Provide space or a formula input box for students to input their answer._)
Explanation:
To find the volume using the method of cylindrical shells, integrate the product of the circumference of a shell, the height of the shell, and the thickness of the shell along the x-axis. The cylindrical shell method uses the following formula:
\[ V = 2\pi \int_{a}^{b} (radius \cdot height \cdot thickness) \, dx \]
Where:
- The radius of the shell is the distance from the axis of rotation (y-axis), which is \( x \).
- The height of the shell is the function \( y = \frac{7}{x} \).
- The thickness is \( dx \).
Thus, the volume becomes:
\[ V = 2\pi \int_{1}^{6} x \left(\frac{7}{x}\right) dx \]
Simplifying the integrand:
\[ V = 2\pi \int_{1}^{6} 7 \, dx \]
\[ V = 2\pi \left[ 7x \right]_{1}^{6} \]
Evaluating the definite integral:
\[ V = 2\pi \left[ 7(6) - 7(1) \right] \]
\[ V = 2\pi \left[ 42 - 7 \right] \]
\[ V = 2\pi (35) \]
\[ V = 70\pi \]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 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,
