Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. xy = 3, x = 0, y = 3, y = 5
Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. xy = 3, x = 0, y = 3, y = 5
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Topic: Calculating Volume Using the Method of Cylindrical Shells**
**Problem Statement:**
Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.
Given curves:
- \(xy = 3\)
- \(x = 0\)
- \(y = 3\)
- \(y = 5\)
**Explanation:**
The problem involves finding the volume of a solid of revolution. The specified region, bounded by the curves, is rotated around the x-axis. To solve this problem, the method of cylindrical shells will be utilized.
The method of cylindrical shells involves:
- Identifying the shell radius and height as functions of y
- Setting up the integral with respect to y
- Evaluating the definite integral over the specified interval \([y = 3, y = 5]\)
**Detailed Steps:**
1. **Determine the functions for integration:** Analyze the provided equations to understand how they bound the region of interest.
2. **Set up the integral:** The formula for the volume \(V\) using cylindrical shells is:
\[
V = 2\pi \int_{c}^{d} (shell \, radius) \times (shell \, height) \, dy
\]
3. **Evaluation:** Substitute and evaluate the integral to find the volume of the solid.
**Additional Tips:**
- Sketch the region and axis of rotation to visualize the problem.
- Check calculus resources for similar problems and examples to reinforce understanding.
This exercise provides a practical application of integration techniques and emphasizes the geometric interpretation of solutions in calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97aba9c2-7733-43fc-8cda-64b1db1637bb%2F579f4601-8c08-4635-a98a-6c718f7895e4%2F29dx5wq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Calculating Volume Using the Method of Cylindrical Shells**
**Problem Statement:**
Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.
Given curves:
- \(xy = 3\)
- \(x = 0\)
- \(y = 3\)
- \(y = 5\)
**Explanation:**
The problem involves finding the volume of a solid of revolution. The specified region, bounded by the curves, is rotated around the x-axis. To solve this problem, the method of cylindrical shells will be utilized.
The method of cylindrical shells involves:
- Identifying the shell radius and height as functions of y
- Setting up the integral with respect to y
- Evaluating the definite integral over the specified interval \([y = 3, y = 5]\)
**Detailed Steps:**
1. **Determine the functions for integration:** Analyze the provided equations to understand how they bound the region of interest.
2. **Set up the integral:** The formula for the volume \(V\) using cylindrical shells is:
\[
V = 2\pi \int_{c}^{d} (shell \, radius) \times (shell \, height) \, dy
\]
3. **Evaluation:** Substitute and evaluate the integral to find the volume of the solid.
**Additional Tips:**
- Sketch the region and axis of rotation to visualize the problem.
- Check calculus resources for similar problems and examples to reinforce understanding.
This exercise provides a practical application of integration techniques and emphasizes the geometric interpretation of solutions in calculus.
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