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 y-axis. y = √x 3.5 3 2.5- y 2 1.5 0.5- 2 4 6
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 y-axis. y = √x 3.5 3 2.5- y 2 1.5 0.5- 2 4 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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7.3
![### Volume of Solids Using the Shell Method
**Problem Statement:**
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 y-axis.
Equation of the curve:
\[ y = \sqrt{x} \]
**Graph Explanation:**
The graph displays a curve represented by the equation \( y = \sqrt{x} \) on the coordinate plane, specifically in the first quadrant. The region between the curve and the x-axis is shaded in blue, illustrating the area of interest that will be revolved around the y-axis to form a solid.
- The x-axis ranges from 0 to approximately 8.
- The y-axis ranges from 0 to 3.5.
- An indicative rectangular strip within the region, highlighted in yellow, exemplifies a shell that will be used in setting up the integral for the volume calculation.
**Shell Method Overview:**
The shell method involves considering a vertical strip parallel to the axis of revolution (y-axis in this case). The strip, when revolved, describes a cylindrical shell. The volume of the solid of revolution is found by integrating the surface area of these cylindrical shells from the start to end of the region's x-values.
**Steps to Solve:**
1. **Identify the limits of integration** based on the x-values where the curve exists.
2. **Determine the formula for the shell's radius and height.**
- Radius: The distance from the y-axis to the strip, \( x \).
- Height: The function \( y = \sqrt{x} \).
3. **Set up the integral for the volume of the solid:**
\[
V = 2\pi \int_{a}^{b} (\text{radius}) \times (\text{height}) \, dx = 2\pi \int_{a}^{b} x \cdot \sqrt{x} \, dx
\]
4. **Evaluate the definite integral** to find the volume.
This process will give the volume of the solid formed by revolving the specified region around the y-axis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe35298e5-10ed-452c-b18c-35d9ef29ce25%2Fe68b37c4-c180-4f9c-a457-4852bf3512df%2Fr0wl5b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Volume of Solids Using the Shell Method
**Problem Statement:**
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 y-axis.
Equation of the curve:
\[ y = \sqrt{x} \]
**Graph Explanation:**
The graph displays a curve represented by the equation \( y = \sqrt{x} \) on the coordinate plane, specifically in the first quadrant. The region between the curve and the x-axis is shaded in blue, illustrating the area of interest that will be revolved around the y-axis to form a solid.
- The x-axis ranges from 0 to approximately 8.
- The y-axis ranges from 0 to 3.5.
- An indicative rectangular strip within the region, highlighted in yellow, exemplifies a shell that will be used in setting up the integral for the volume calculation.
**Shell Method Overview:**
The shell method involves considering a vertical strip parallel to the axis of revolution (y-axis in this case). The strip, when revolved, describes a cylindrical shell. The volume of the solid of revolution is found by integrating the surface area of these cylindrical shells from the start to end of the region's x-values.
**Steps to Solve:**
1. **Identify the limits of integration** based on the x-values where the curve exists.
2. **Determine the formula for the shell's radius and height.**
- Radius: The distance from the y-axis to the strip, \( x \).
- Height: The function \( y = \sqrt{x} \).
3. **Set up the integral for the volume of the solid:**
\[
V = 2\pi \int_{a}^{b} (\text{radius}) \times (\text{height}) \, dx = 2\pi \int_{a}^{b} x \cdot \sqrt{x} \, dx
\]
4. **Evaluate the definite integral** to find the volume.
This process will give the volume of the solid formed by revolving the specified region around the y-axis.
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