3. Let R be the region bounded by y = ln x, the x-axis, and the line x = e. Find the volume of the solid of revolution obtained by revolving R about the x-axis. Be sure to sketch the closed region R and the resulting solid.
3. Let R be the region bounded by y = ln x, the x-axis, and the line x = e. Find the volume of the solid of revolution obtained by revolving R about the x-axis. Be sure to sketch the closed region R and the resulting solid.
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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Question
![3. Let \( R \) be the region bounded by \( y = \ln x \), the \( x \)-axis, and the line \( x = e \). Find the volume of the solid of revolution obtained by revolving \( R \) about the \( x \)-axis. Be sure to sketch the closed region \( R \) and the resulting solid.
**Explanation of the Problem:**
1. **Region \( R \):**
- **Boundaries:**
- The curve \( y = \ln x \).
- The \( x \)-axis which is the line \( y = 0 \).
- The vertical line \( x = e \).
2. **Solid of Revolution:**
- The region \( R \) is revolved around the \( x \)-axis to form a solid.
- The task is to find the volume of this solid.
3. **Sketching:**
- **The Region:**
- Plot \( y = \ln x \) from \( x = 1 \) to \( x = e \).
- Mark the area under the curve down to the \( x \)-axis and up to \( x = e \).
- **Resulting Solid:**
- Visualize the rotation of this region 360 degrees around the \( x \)-axis, creating a three-dimensional shape.
**Mathematical Approach:**
- Use the disk method to calculate the volume:
\[
V = \pi \int_{1}^{e} (\ln x)^2 \, dx
\]
- Evaluate the integral to find the exact volume of the solid.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F29f9c049-e4a2-4005-a5d6-aa9835638972%2F206a1ecf-3064-4dcf-92f2-ff6d2e10eebc%2Fnulkaps_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Let \( R \) be the region bounded by \( y = \ln x \), the \( x \)-axis, and the line \( x = e \). Find the volume of the solid of revolution obtained by revolving \( R \) about the \( x \)-axis. Be sure to sketch the closed region \( R \) and the resulting solid.
**Explanation of the Problem:**
1. **Region \( R \):**
- **Boundaries:**
- The curve \( y = \ln x \).
- The \( x \)-axis which is the line \( y = 0 \).
- The vertical line \( x = e \).
2. **Solid of Revolution:**
- The region \( R \) is revolved around the \( x \)-axis to form a solid.
- The task is to find the volume of this solid.
3. **Sketching:**
- **The Region:**
- Plot \( y = \ln x \) from \( x = 1 \) to \( x = e \).
- Mark the area under the curve down to the \( x \)-axis and up to \( x = e \).
- **Resulting Solid:**
- Visualize the rotation of this region 360 degrees around the \( x \)-axis, creating a three-dimensional shape.
**Mathematical Approach:**
- Use the disk method to calculate the volume:
\[
V = \pi \int_{1}^{e} (\ln x)^2 \, dx
\]
- Evaluate the integral to find the exact volume of the solid.
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