Use the disk method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = ex, y = 0, x = 0, X=6
Use the disk method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = ex, y = 0, x = 0, 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
Related questions
Question
I need help with these two problems please help me
I know its suppose to be only 1 question but I have no more question left to help me
![---
### Finding the Volume of a Solid Using the Disk Method
In this exercise, we aim to determine the volume of the solid generated by revolving the region bounded by the given graphs around the x-axis. The equations defining the region are:
\[ y = e^{-x} \]
\[ y = 0 \]
\[ x = 0 \]
\[ x = 6 \]
#### Steps for Using the Disk Method
1. **Identify the Bounded Region:**
- The region is bounded by the curve \( y = e^{-x} \) from \( x = 0 \) to \( x = 6 \) and by the lines \( y = 0 \) (the x-axis), \( x = 0 \) (the y-axis), and \( x = 6 \) (a vertical line).
2. **Set Up the Integral:**
- To use the disk method, we revolve the region around the x-axis, generating disks with radii equal to \( y = e^{-x} \). The volume of each infinitesimally small disk is \(\pi y^2 \, dx\).
3. **Integral Formulation:**
- The volume \( V \) is given by:
\[
V = \pi \int_{0}^{6} \left( e^{-x} \right)^2 \, dx
\]
4. **Simplify and Compute the Integral:**
- Simplify the integrand:
\[
\left( e^{-x} \right)^2 = e^{-2x}
\]
- The volume integral becomes:
\[
V = \pi \int_{0}^{6} e^{-2x} \, dx
\]
#### Diagram Explanation
- **Graph:**
- The provided graph shows an exponential decay function \( y = e^{-x} \), a horizontal line at \( y = 0 \), and vertical lines at \( x = 0 \) and \( x = 6 \).
- The shaded area between \( x = 0 \) and \( x = 6 \) under the curve \( y = e^{-x} \) is revolved around the x-axis to create the solid.
Using these steps, you can compute the volume of the solid generated.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00adfe79-2c61-4bb3-a45f-f8f9088ebf01%2Fe2ddc45e-674b-4e38-9d11-e2c7bddbd880%2Fng4wwmd_processed.png&w=3840&q=75)
Transcribed Image Text:---
### Finding the Volume of a Solid Using the Disk Method
In this exercise, we aim to determine the volume of the solid generated by revolving the region bounded by the given graphs around the x-axis. The equations defining the region are:
\[ y = e^{-x} \]
\[ y = 0 \]
\[ x = 0 \]
\[ x = 6 \]
#### Steps for Using the Disk Method
1. **Identify the Bounded Region:**
- The region is bounded by the curve \( y = e^{-x} \) from \( x = 0 \) to \( x = 6 \) and by the lines \( y = 0 \) (the x-axis), \( x = 0 \) (the y-axis), and \( x = 6 \) (a vertical line).
2. **Set Up the Integral:**
- To use the disk method, we revolve the region around the x-axis, generating disks with radii equal to \( y = e^{-x} \). The volume of each infinitesimally small disk is \(\pi y^2 \, dx\).
3. **Integral Formulation:**
- The volume \( V \) is given by:
\[
V = \pi \int_{0}^{6} \left( e^{-x} \right)^2 \, dx
\]
4. **Simplify and Compute the Integral:**
- Simplify the integrand:
\[
\left( e^{-x} \right)^2 = e^{-2x}
\]
- The volume integral becomes:
\[
V = \pi \int_{0}^{6} e^{-2x} \, dx
\]
#### Diagram Explanation
- **Graph:**
- The provided graph shows an exponential decay function \( y = e^{-x} \), a horizontal line at \( y = 0 \), and vertical lines at \( x = 0 \) and \( x = 6 \).
- The shaded area between \( x = 0 \) and \( x = 6 \) under the curve \( y = e^{-x} \) is revolved around the x-axis to create the solid.
Using these steps, you can compute the volume of the solid generated.
![### Calculus: Volume of Solids of Revolution Using the Shell Method
**Problem Statement:**
Use the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis:
\[ y = \frac{1}{x^2}, \quad y = 0, \quad x = 5, \quad \textcolor{red}{x = 8} \]
### Explanation:
To solve for the volume of the solid generated by revolving the given region about the y-axis using the shell method, follow these steps:
1. **Identify the region of integration:**
- The region is bounded by the function \( y = \frac{1}{x^2} \), the x-axis \( y = 0 \), and the vertical lines \( x = 5 \) and \( x = 8 \).
2. **Set up the volume integral using the shell method:**
- The shell method involves integrating along the axis perpendicular to the axis of rotation (in this case, the x-axis).
- The radius of the shell at a point \( x \) is the distance from the y-axis, which is \( x \).
- The height of the shell is given by the function, \( \frac{1}{x^2} \).
- The thickness of each shell is \( dx \).
3. **Formulate the integral:**
- The volume \( V \) is given by the integral:
\[
V = \int_{a}^{b} 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx
\]
- In this case, \( a = 5 \), \( b = 8 \), radius \( = x \), height \( = \frac{1}{x^2} \):
\[
V = \int_{5}^{8} 2\pi x \cdot \frac{1}{x^2} \, dx
\]
4. **Simplify and evaluate the integral:**
- The integrand simplifies to:
\[
V = \int_{5}^{8} 2\pi \cdot \frac{1}{x} \, dx
\]
- This can be evaluated as:
\[
V = 2\pi \int_{5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00adfe79-2c61-4bb3-a45f-f8f9088ebf01%2Fe2ddc45e-674b-4e38-9d11-e2c7bddbd880%2Fsxr7npd_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus: Volume of Solids of Revolution Using the Shell Method
**Problem Statement:**
Use the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis:
\[ y = \frac{1}{x^2}, \quad y = 0, \quad x = 5, \quad \textcolor{red}{x = 8} \]
### Explanation:
To solve for the volume of the solid generated by revolving the given region about the y-axis using the shell method, follow these steps:
1. **Identify the region of integration:**
- The region is bounded by the function \( y = \frac{1}{x^2} \), the x-axis \( y = 0 \), and the vertical lines \( x = 5 \) and \( x = 8 \).
2. **Set up the volume integral using the shell method:**
- The shell method involves integrating along the axis perpendicular to the axis of rotation (in this case, the x-axis).
- The radius of the shell at a point \( x \) is the distance from the y-axis, which is \( x \).
- The height of the shell is given by the function, \( \frac{1}{x^2} \).
- The thickness of each shell is \( dx \).
3. **Formulate the integral:**
- The volume \( V \) is given by the integral:
\[
V = \int_{a}^{b} 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx
\]
- In this case, \( a = 5 \), \( b = 8 \), radius \( = x \), height \( = \frac{1}{x^2} \):
\[
V = \int_{5}^{8} 2\pi x \cdot \frac{1}{x^2} \, dx
\]
4. **Simplify and evaluate the integral:**
- The integrand simplifies to:
\[
V = \int_{5}^{8} 2\pi \cdot \frac{1}{x} \, dx
\]
- This can be evaluated as:
\[
V = 2\pi \int_{5
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 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,
