Integration
Why do we need to study Integration?
by M. Bourne
The Petronas Towers
of Kuala Lumpur
Often we know the relationship involving the rate of change of two variables, but we may need to know the direct relationship between the two variables. For example, we may know the velocity of an object at a particular time, but we may want to know the position of the object at that time.
To find this direct relationship, we need to use the process which is opposite to differentiation. This is called integration (or antidifferentiation).
The processes of integration are used in many applications.
The Petronas Towers in Kuala Lumpur experience high forces due to winds. Integration was used to design the building for strength.
Sydney Opera House
The Sydney Opera House is a very unusual design based on slices out of a ball. Many differential equations (one type of integration) were solved in the design of this building.
Wine cask
Historically, one of the first uses of integration was in finding the volumes of wine-casks (which have a curved surface).
We learn to find the volume of these objects later (in volume of solid of revolution).
Other uses of integration include finding areas under curved surfaces, centres of mass, displacement and velocity, fluid flow, modelling the behaviour of objects under stress, etc.
In this Chapter
Related Sections in Interactive Mathematics
The Introduction to Calculus has a brief history of calculus.
Methods of Integration, which shows more advanced techniques for integrating.
Applications of Integration, which demonstrates how to solve several problems using integration.
Fourier Series and Fast Fourier Transform show the use of integration in the analysis of electronics.
Differential Equations are a special type of integration.
Laplace Transform of an integral is an important process in electronics.
- 1. The Differential
- 2. Antiderivatives and The Indefinite Integral - shows how to find simple integrals
- 3. The Area Under a Curve - shows how to use integration to find the area under a curve
- 4. The Definite Integral - substitution method and some applications
- 5. Trapezoidal Rule - numerical integration, when we cannot find an expressions
- 6. Simpsonâ€™s Rule - a better method for integrating numerically
- Integration Mini-lectures - some Scientific Notebook-based lectures on integration basics
- Millionaire Calculus Game
We begin with a discussion of the differential, because it involves some of the concepts and notation used in the study of integration. The Differential »