# Applications of Integration

by M. Bourne

Chapter Introduction

In this chapter we learn how to find displacement (from velocity) and velocity (from acceleration) using the indefinite integral. There are also some **electronics** applications in this section.

In primary school, we learned how to find areas of shapes with straight sides (e.g. area of a triangle or rectangle). But how do you find **areas when the sides are curved**?

The Volume of Solid of Revolution section explains how to use integration to find the volume of an object with curved sides, e.g. wine barrels.

We see how to use integration to find the centroid of an area with curved sides.

The Moments of Inertia section explains how to find the resistance of a rotating body. We use integration when the shape has curved sides.

Work by a Variable Force shows how to find the work done on an object when the force is not constant. This section includes **Hooke's Law** for springs.

Other applications in this chapter include:

- Electric Charges
- Average Valueof a function
- Head Injury Criterion, an application of average value and used in road safety research.
- Force by Liquid Pressure

In each case, we solve the problem by considering the simple case first. Usually this means the area or volume has straight sides. Then we extend the straight-sided case to consider curved sides. We need to use** integration** because we have curved sides and cannot use the simple formulas any more.

The chapter begins with 1. Applications of the Indefinite Integral »

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Chapter Contents

- Applications of Integration
- 1. Applications of the Indefinite Integral
- 2. Area Under a Curve by Integration
- 3. Area Between 2 Curves using Integration
- 4. Volume of Solid of Revolution by Integration
- 5. Centroid of an Area by Integration
- 6. Moments of Inertia by Integration
- 7. Work by a Variable Force using Integration
- 8. Electric Charges by Integration
- 9. Average Value of a Function by Integration
- Head Injury Criterion (HIC) pt 1: Severity Index
- Head Injury Criterion (HIC) pt 2: HIC Index, example
- 10. Force Due to Liquid Pressure by Integration
- 11. Arc Length of a Curve using Integration
- 12. Arc Length of Curve: Parametric, Polar Coordinates