# 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.

Continue reading below ⇩

Wine cask

Historically, one of the first uses of integration was in finding the **volumes of wine-casks** (which have a curved surface).

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.

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 »

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Chapter Contents

- Integration
- 1. The Differential
- 2. Antiderivatives and The Indefinite Integral
- 3. The Area Under a Curve
- 4. The Definite Integral
- 5. Trapezoidal Rule
- 6. Simpson’s Rule
- 6a. Riemann Sums
- 6b. Fundamental Theorem of Calculus Applet
- 7. Integration Mini-lectures
- 7a. The Differential
- 7b. Difference Between Differentiation and Integration
- 7c. Given dy/dx, find y = f(x)
- 7d. Integration by Substitution
- 7e. Difference Between Definite and Indefinite Integrals
- 7f. Area Under a Curve
- 8. Millionaire Calculus Game