Volumes of Revolution

Emily Davis
7 min read
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Study Guide Overview
This study guide covers the washer method for finding the volume of revolution around axes other than the x or y-axis. It explains how to set up and solve integrals for rotations around lines parallel to both the x-axis and y-axis. Examples, practice questions, and exam strategies are included.
#Volume with Washer Method Revolving Around Other Axes
#Table of Contents
- Introduction
- Volume of Revolution Around a Line Parallel to the x-axis
- Volume of Revolution Around a Line Parallel to the y-axis
- Practice Questions
- Glossary
- Summary and Key Takeaways
- Exam Strategy
#Introduction
The washer method is a technique used to find the volume of a solid of revolution when the solid is generated by rotating a region bounded by two curves around an axis. This method involves integrating the area of washers (or disks with holes) formed by the rotation.
#Volume of Revolution Around a Line Parallel to the x-axis
To calculate the volume of revolution around a line parallel to the x-axis using the washer method, follow these steps:
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Identify the functions and the interval: Let and be continuous functions on the interval , with closer to the horizontal line than .
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Set up the integral: The volume of revolution is given by:
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Ensure proper boundaries: If the curves swap places over the interval, split the calculation into separate integrals.
#Worked Example
Let be the region enclosed by the graphs of and . The region is rotated about the horizontal line .
Step-by-Step Solution:
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Find the points of intersection: So, and .
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Identify the functions closest to : is closer to than . Thus, and .
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Set up and solve the integral:
#Volume of Revolution Around a Line Parallel to the y-axis
To calculate the volume of revolution around a line parallel to the y-axis, follow these steps:
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Identify the functions and the interval: Let and be continuous functions on the interval , with closer to the vertical line than .
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Set up the integral: The volume of revolution is given by:
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Ensure proper boundaries: If the curves swap places over the interval, split the calculation into separate integrals.
#Worked Example
Let be the region enclosed by the graphs of and . The region is rotated about the vertical line .
Step-by-Step Solution:
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Rewrite the functions as functions of :
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Find the points of intersection: So, and .
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Identify the functions closest to : is closer to than . Thus, and .
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Set up and solve the integral:
#Practice Questions
Practice Question
#Glossary
- Volume of Revolution: The volume of a solid formed by rotating a region around an axis.
- Washer Method: A technique to find the volume of a solid of revolution when the solid has a hole in the middle.
- Integral: A mathematical operation that sums the area under a curve.
- Boundaries: The limits of integration, often determined by the intersection points of curves.
#Summary and Key Takeaways
- The washer method involves integrating the area of washers formed by rotating a region around an axis.
- Ensure the functions are properly identified and the correct integral setup is used.
- Pay attention to the boundaries and whether the curves swap places within the interval.
- Practice solving integrals to become proficient in applying the washer method.
#Exam Strategy
- Read the question carefully: Identify the functions and the axis of rotation.
- Sketch the region: Visualize the area being rotated to understand the setup.
- Set up the integral correctly: Use the washer method formula and ensure the limits of integration are accurate.
- Check your work: Verify the points of intersection and the integral setup before solving.
By following these steps and practicing regularly, you'll be well-prepared to tackle volume of revolution problems in your exams.
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