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 ...

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 $y=b$ ?

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

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

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

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