Volumes of Revolution

Emily Davis

5 min read

Next Topic - Disc Method Around the y-Axis

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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∫_a^b[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

#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 $y=f(x)$ (and other boundary equations) is rotated $2\pi$ radians ( $360°$ ) around the $x$ -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 $f$ , if the region bounded by:

the curve $y=f(x)$ and the $x$ -axis
between $x=a$ and $x=b$

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

 $V = \pi \int_...$

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Previous Topic - Semicircles as Cross Sections Next Topic - Disc Method Around the y-Axis

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