19. Find the volume obtained by rotating f(r) = x(1 – x) around the x-axis. Then find the volume obtained by rotating around the y-axis. %3D
19. Find the volume obtained by rotating f(r) = x(1 – x) around the x-axis. Then find the volume obtained by rotating around the y-axis. %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![Certainly! Here's a transcription suitable for an educational website:
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**Problem 19: Calculating Volumes of Revolution**
- **Objective**: Determine the volume of the solid formed by rotating the function \( f(x) = x(1-x) \).
1. **Rotation Around the \( x \)-axis**:
- Use the disk method to find the volume of the solid obtained by rotating the graph of the function \( f(x) = x(1-x) \) around the \( x \)-axis.
2. **Rotation Around the \( y \)-axis**:
- Apply the shell method to calculate the volume of the solid formed by rotating the graph of the function \( f(x) = x(1-x) \) around the \( y \)-axis.
- **Methods to Apply**:
- **Disk Method**: Integrate using \(\pi \int [f(x)]^2 \, dx\).
- **Shell Method**: Integrate using \(2\pi \int x \cdot f(x) \, dx\).
Solve and explain each part in detail, and ensure to write down any intermediate steps to foster understanding.
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Transcribed Image Text:Certainly! Here's a transcription suitable for an educational website:
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**Problem 19: Calculating Volumes of Revolution**
- **Objective**: Determine the volume of the solid formed by rotating the function \( f(x) = x(1-x) \).
1. **Rotation Around the \( x \)-axis**:
- Use the disk method to find the volume of the solid obtained by rotating the graph of the function \( f(x) = x(1-x) \) around the \( x \)-axis.
2. **Rotation Around the \( y \)-axis**:
- Apply the shell method to calculate the volume of the solid formed by rotating the graph of the function \( f(x) = x(1-x) \) around the \( y \)-axis.
- **Methods to Apply**:
- **Disk Method**: Integrate using \(\pi \int [f(x)]^2 \, dx\).
- **Shell Method**: Integrate using \(2\pi \int x \cdot f(x) \, dx\).
Solve and explain each part in detail, and ensure to write down any intermediate steps to foster understanding.
---
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