Volumes of Revolution

Emily Davis

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Next Topic - Washer Method Around Other Axes

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Study Guide Overview

This study guide covers the washer method for finding the volume of revolution around the y-axis. It explains the step-by-step methodology including identifying functions, setting up the integral with respect to y, rewriting functions in terms of y, and determining integration boundaries. It provides a worked example, practice questions, and a glossary of key terms like washer method, volume of revolution, and integral. Key takeaways emphasize rewriting functions, correct integration, and boundary identification.

#Volume with Washer Method Revolving Around the y-axis

#Introduction

The washer method is a technique for calculating the volume of a solid of revolution, particularly when the solid is created by rotating a region around the y-axis. This method is useful when there is a gap between the region to be rotated and the y-axis.

#Key Concepts

Key Concept

Washer Method: A technique for finding the volume of a solid of revolution with a gap between the region and the axis.
Volume of Revolution: The volume of a 3D shape formed by rotating a 2D region around an axis.

#Step-by-Step Methodology

Identify the Functions and Interval:
- Determine the functions $f(y)$ and $g(y)$ such that $|f(y)| < |g(y)|$ on the interval $[a, b]$ .
- Ensure $f(y)$ is closer to the y-axis than $g(y)$ over the interval.
Set Up the Integral:
- If the region is bounded by the curves $x_1 = f(y)$ and $x_2 = g(y)$ between $y = a$ and $y = b$ , the volume of revolution is given by: $V = \pi \int_{a}^{b} \left( x_2^2 - x_1^2 \right) , dy$
- Note that the integration is done with respect to $y$ .
Rewrite Functions if Necessary:
- If the functions are given as $y = p(x)$ and $y = q(x)$ , rewrite them as functions of $y$ .
Ensure Correct Boundaries:
- If $y = a$ and $y = b$ are not stated, determine these boundaries based on the x-axis ( $y = 0$ ) or the points of intersection of the curves.
Perform the Integration:
- Integrate the function $\pi \left( x_2^2 - x_1^2 \right)$ with respect to $y$ over the interval $[a, b]$ .

### Example Let

R

be the region enclosed by the graphs of

f(x) = \frac{1}{4}x^2

and

g(x) = x

Solution:

Rewrite Functions:
- For $y = f(x)$ : $y = \frac{1}{4}x^2 \implies x^2 = 4y \implies x = 2\sqrt{y}$
- For $y = g(x)$ : $y = x \implies x = y$
Identify Intersection Points:
- Set $2\sqrt{y} = y$ : $y^2 = 4y \implies y(y - 4) = 0 \implies y = 0 \text{ or } y = 4$
- Thus, $a = 0$ and $b = 4$ .
Set Up Integral: $V = \pi \int_{0}^{4} \left( (2\sqrt{y})^2 - (y)^2 \right) , dy = \pi \int_{0}^{4} \left( 4y - y^2 \right) , dy$
Solve Integral: $V = \pi \left[ 2y^2 - \frac{1}{3}y^3 \right]_{0}^{4} = \pi \left( \left( 2(4)^2 - \frac{1}{3}(4)^3 \right) - 0 \right) = \frac{32\pi}{3} \approx 33.510 , \text{units}^3$

Exam Tip

#Exam Tip

Be careful not to confuse $\left( x_2^2 - x_1^2 \right)$ with $\left( x_2 - x_1 \right)^2$ . These are not equal: $\left( x_2 - x_1 \right)^2 = x_2^2 - 2x_1 x_2 + x_1^2$

#Practice Questions

Practice Question

Find the volume of the solid formed by rotating the region between $y = x^2$ and $y = 2x$ around the y-axis from $y = 0$ to $y = 4$ .

Practice Question

Calculate the volume of the solid formed by rotating the region between $y = 3 - x$ and $y = x^2 - 1$ around the y-axis from $y = -1$ to $y = 3$ .

#Glossary

- **Washer Method:** A method to calculate volumes of solids of revolution with a gap between the region and the axis. - **Volume of Revolution:** The volume formed when a region is rotated around an axis. - **Integral:** A mathematical operation used to calculate areas, volumes, and other quantities.

#Summary and Key Takeaways

The washer method is a powerful technique for calculating the volume of a solid of revolution around the y-axis, especially when there is a gap between the region and the axis. Key steps include identifying the functions and interval, setting up the integral, rewriting functions if necessary, ensuring correct boundaries, and performing the integration.

Key Takeaways:

Always rewrite functions in terms of $y$ if given in terms of $x$ .
Ensure integration is done with respect to $y$ .
Be mindful of the boundaries and points of intersection.
Double-check the integral setup to avoid common mistakes.

By following these guidelines, students can effectively utilize the washer method to solve problems related to volumes of revolution around the y-axis.

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Question 1 of 9

Ready to find some volumes? 🚀 What is the general formula for the volume of a solid of revolution around the y-axis using the Washer Method, where $x_1(y)$ and $x_2(y)$ are the inner and outer radii, respectively, and the region is bounded by $y=a$ and $y=b$ ?

$V = \pi \int_{a}^{b} (x_2(y) - x_1(y))^2 , dy$

$V = \pi \int_{a}^{b} (x_2^2(y) - x_1^2(y)) , dy$

$V = \int_{a}^{b} (x_2^2(y) - x_1^2(y)) , dy$

$V = 2\pi \int_{a}^{b} y(x_2(y) - x_1(y)) , dy$