Book contents
- Frontmatter
- Contents
- Preface
- 1 Basic functions
- 2 Functions for science 1: the exponential function
- 3 Functions for science 2: trigonometric functions
- 4 Functions for science 3: inverse functions
- 5 Other functions of science
- 6 Differentiation 1: rates of change
- 7 Differentiation 2: stationary points
- 8 Differentiation 3: approximation of functions
- 9 Integration 1: introduction and standard forms
- 10 Integration 2: techniques of integration
- 11 Integration 3: further techniques
- 12 First-order ordinary differential equations
- 13 Second-order ordinary differential equations
- 14 Statistics 1: frequency distributions and associated measures
- 15 Statistics 2: probability and probability distributions
- 16 Statistics 3: sampling, sampling distributions and hypothesis testing
- 17 Partial differentiation 1: introduction
- 18 Partial differentiation 2: stationary points
- Answers to the exercises
- Index
9 - Integration 1: introduction and standard forms
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- 1 Basic functions
- 2 Functions for science 1: the exponential function
- 3 Functions for science 2: trigonometric functions
- 4 Functions for science 3: inverse functions
- 5 Other functions of science
- 6 Differentiation 1: rates of change
- 7 Differentiation 2: stationary points
- 8 Differentiation 3: approximation of functions
- 9 Integration 1: introduction and standard forms
- 10 Integration 2: techniques of integration
- 11 Integration 3: further techniques
- 12 First-order ordinary differential equations
- 13 Second-order ordinary differential equations
- 14 Statistics 1: frequency distributions and associated measures
- 15 Statistics 2: probability and probability distributions
- 16 Statistics 3: sampling, sampling distributions and hypothesis testing
- 17 Partial differentiation 1: introduction
- 18 Partial differentiation 2: stationary points
- Answers to the exercises
- Index
Summary
We now begin three chapters about integration.
As the word implies, integration is about bringing things together. In mathematics the process of integration involves bringing together function values in a special way to form a summation. This sum of function values is called an integral and we shall see that there are many circumstances in which we calculate the value of some quantity by the process of integration. At first sight it will appear that integration is just the opposite of differentiation, i.e. in a sense what we appear to be doing is ‘undoing’ the differential of a function. This process is called anti-differentiation. For example, cos (x) is the derivative of sin (x), and sin (x) is the anti-derivative of cos (x).
However, integration and antidifferentiation are conceptually different. The fact that the two processes give the same answer is a consequence of an important theorem called the Fundamental Theorem of Calculus. In this first chapter we define what we mean by the integral of a function and investigate the results of integrating the basic functions we have met before.
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- Chapter
- Information
- Introductory Mathematics through Science Applications , pp. 233 - 258Publisher: Cambridge University PressPrint publication year: 1989