### Introduction

In the previous Section we saw how to evaluate double integrals over simple rectangular regions. We now see how to extend this to non-rectangular regions.

In this Section we introduce functions as the limits of integration, these functions define the region over which the integration is performed. These regions can be non-rectangular. Extra care now must be taken when changing the order of integration. Producing a sketch of the region is often very helpful.

#### Prerequisites

- have a thorough understanding of the various techniques of integration
- be familiar with the concept of a function of two variables
- have completed Section 27.1
- be able to sketch a function in the plane

#### Learning Outcomes

- evaluate double integrals over non-rectangular regions

#### Contents

1 Functions as limits of integration1.1 Splitting the region of integration

2 Order of integration

3 Evaluating surface integrals using polar coordinates

3.1 Polar coordinates

3.2 Finding surface integrals with polar coordinates

4 Applications of surface integration

4.1 Force on a dam

4.2 Centre of pressure

5 Engineering Example 1

5.1 Volume of liquid in an elliptic tank