Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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 line x = 5.
![### Calculating Volume using the Shell Method
To find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line \( x = 5 \), we will use the shell method.
The equations that bound the region are:
- \( y = \sqrt{x} \)
- \( y = 0 \)
- \( x = 4 \)
The shell method integrates with respect to the y-axis, creating cylindrical shells. Here's the detailed procedure:
1. **Sketch the Region**: Identify the region bounded by \( y = \sqrt{x} \), \( y = 0 \), and \( x = 4 \).
2. **Set Up the Shell Radius and Height**:
- The radius of a shell at any point \( x \) is \( 5 - x \).
- The height of a shell at any point \( x \) is \( \sqrt{x} - 0 = \sqrt{x} \).
3. **Integration Limits**: Since \( x \) ranges from 0 to 4, these will be our limits of integration.
4. **Volume Integral**:
\[
V = \int_{0}^{4} 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx
\]
\[
V = \int_{0}^{4} 2\pi \cdot (5 - x) \cdot \sqrt{x} \, dx
\]
5. **Evaluate the Integral**:
\[
V = 2\pi \int_{0}^{4} (5 - x) \sqrt{x} \, dx
\]
Split the integral for easier calculation:
\[
V = 2\pi \left( \int_{0}^{4} 5\sqrt{x} \, dx - \int_{0}^{4} x\sqrt{x} \, dx \right)
\]
Evaluate each part:
\[
\int_{0}^{4} 5\sqrt{x} \, dx = 5 \cdot \int_{0}^{4} x^{1/2} \, dx = 5 \cdot \left[ \frac{2}{3} x^{3/2} \right]_{0}^{4} = 5 \cdot \frac{](https://content.bartleby.com/qna-images/question/da6dd950-15a2-4e47-84e9-a8f156faff1f/9a3db690-fe11-460c-95b9-9a0596d788ee/5bo228_processed.png)
Transcribed Image Text:### Calculating Volume using the Shell Method
To find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line \( x = 5 \), we will use the shell method.
The equations that bound the region are:
- \( y = \sqrt{x} \)
- \( y = 0 \)
- \( x = 4 \)
The shell method integrates with respect to the y-axis, creating cylindrical shells. Here's the detailed procedure:
1. **Sketch the Region**: Identify the region bounded by \( y = \sqrt{x} \), \( y = 0 \), and \( x = 4 \).
2. **Set Up the Shell Radius and Height**:
- The radius of a shell at any point \( x \) is \( 5 - x \).
- The height of a shell at any point \( x \) is \( \sqrt{x} - 0 = \sqrt{x} \).
3. **Integration Limits**: Since \( x \) ranges from 0 to 4, these will be our limits of integration.
4. **Volume Integral**:
\[
V = \int_{0}^{4} 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx
\]
\[
V = \int_{0}^{4} 2\pi \cdot (5 - x) \cdot \sqrt{x} \, dx
\]
5. **Evaluate the Integral**:
\[
V = 2\pi \int_{0}^{4} (5 - x) \sqrt{x} \, dx
\]
Split the integral for easier calculation:
\[
V = 2\pi \left( \int_{0}^{4} 5\sqrt{x} \, dx - \int_{0}^{4} x\sqrt{x} \, dx \right)
\]
Evaluate each part:
\[
\int_{0}^{4} 5\sqrt{x} \, dx = 5 \cdot \int_{0}^{4} x^{1/2} \, dx = 5 \cdot \left[ \frac{2}{3} x^{3/2} \right]_{0}^{4} = 5 \cdot \frac{
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