#Volume with Washer Method Revolving Around Other Axes

#Table of Contents

1. Introduction
2. Volume of Revolution Around a Line Parallel to the x-axis
   • Worked Example
3. Volume of Revolution Around a Line Parallel to the y-axis
   • Worked Example
4. Practice Questions
5. Glossary
6. Summary and Key Takeaways
7. 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:

1. Identify the functions and the interval: Let $f(x)$ and $g(x)$ be continuous functions on the interval $[a, b]$, with $f(x)$ closer to the horizontal line $y = k$ than $g(x)$.

2. Set up the integral: The volume of revolution $V$ is given by: $$V = \pi \int_{a}^{b} \left[ (g(x) - k)^2 - (f(x) - k)^2 \right] , dx$$

3. Ensure proper boundaries: If the curves swap places over the interval, split the calculation into separate integrals.