#Volume with Washer Method Revolving Around the x-axis

#Table of Contents

1. Introduction
2. When to Use the Washer Method
3. Calculating Volume Using the Washer Method
4. Exam Tips
5. Worked Example
6. Practice Questions
7. Glossary
8. Summary and Key Takeaways

#Introduction

The washer method is a powerful tool for calculating the volume of a solid of revolution when there is a gap between the region being rotated and the axis of rotation. This method extends the disk method by accounting for the inner radius, which is subtracted from the outer radius.

#When to Use the Washer Method

The washer method should be used when there is a gap between the region being rotated and the axis of rotation (x-axis in this case).

#Cross Section

For $x$ between $a$ and $b$, the cross-section of the solid of revolution will have the shape of a washer, with: $$\text{Area} = \pi \left( \left( g(x) \right)^2 - \left( f(x) \right)^2 \right)$$

#Calculating Volume Using the Washer Method

To calculate the volume of revolution around the x-axis using the washer method, follow these steps:

1. Identify the Curves: Let $f$ and $g$ be continuous functions such that $|f(x)| < |g(x)|$ on the interval $[a, b]$...