### Introduction

Often in Engineering it is necessary to find the sum of a quantity over an area or surface. This can be achieved by means of a surface integral also known as a double integral i.e. a function is integrated twice, once with respect to one variable and subsequently with respect to another variable. This Section looks at the concept of the double integral and how to evaluate a double integral over a rectangular area.

#### Prerequisites

- thoroughly understand the various techniques of integration
- be familiar with the concept of a function of two variables

#### Learning Outcomes

- understand the concept of a surface integral
- integrate a function over a rectangular region

#### Contents

1 An example of a surface integral2 Single and double integrals

3 ‘Inner’ and ‘Outer’ integrals

4 Integration over rectangular areas

5 Special cases

6 Applications of surface integration over rectangular areas

6.1 Force on a dam

6.2 Centre of pressure

6.3 Areas and moments