### Introduction

In the previous two Sections, surface integrals (or double integrals) were introduced i.e. functions were integrated with respect to one variable and then with respect to another variable. It is often useful in engineering to extend the process to an integration with respect to three variables i.e. a volume integral or triple integral. Many of the processes and techniques involved in double integration are relevant to triple integration.

#### Prerequisites

- have a thorough understanding of the various techniques of integration
- be familiar with the concept of a function of two variables
- have studied Sections 27.1 and 27.2 on double integration
- be able to visualise or sketch a function in three variables.

#### Learning Outcomes

- evaluate triple integrals

#### Contents

1 Example of volume integral: mass of water in a reservoir2 Evaluating triple integrals

3 Higher order integrals

4 Applications of triple and higher integrals

4.1 Volume

4.2 Mass

4.3 Mass of water in a reservoir

4.4 Centre of mass

4.5 Moment of inertia

5 Engineering Example 2

5.1 Radioactive decay