zuai-logo

Volumes of Revolution

Emily Davis

Emily Davis

5 min read

Listen to this study note

Study Guide Overview

This study guide covers calculating the volume of revolution around the x-axis using the disc method. It explains the formula V = π∫ab[f(x)]² dx and provides a worked example, practice questions, exam tips, and a glossary of key terms like solid of revolution, region, and definite integral.

Volume with Disc Method Revolving Around the x-axis

Table of Contents

  1. Introduction to Volume of Revolution
  2. Using the Disc Method
  3. Exam Tips
  4. Worked Example
  5. Practice Questions
  6. Summary and Key Takeaways
  7. Glossary

Introduction to Volume of Revolution

What is a Volume of Revolution Around the x-axis?

A solid of revolution is formed when an area bounded by a function y=f(x)y=f(x) (and other boundary equations) is rotated 2\pi radians (360°) around the xx-axis. The volume of revolution is the volume of this solid.

Be careful – the 'front' and 'back' of this solid are flat because they were created from straight (vertical) lines. 3D sketches can be misleading.

Using the Disc Method

How to Calculate a Volume of Revolution Around the x-axis?

For a continuous function ff, if the region bounded by:

  • the curve y=f(x)y=f(x) and the xx-axis
  • between x=ax=a and x=bx=b

is rotated 2\pi radians (360°) around the xx-axis, then the volume of revolution is:

V = \pi \int_...

Question 1 of 8

What is a solid of revolution? 🤔

A 2D shape rotated around an axis

A 3D shape formed by rotating an area around an axis

A 3D shape formed by shifting an area

Any 3D shape with circular cross-sections