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