3.3 Integration: Introduction to Notation
In this section, you will practise reading out mathematical symbols. This is important because your lecturers will often read mathematical statements in lectures, and you will also need to communicate about mathematics with others.
Mathematics uses two kinds of systems of communication:
- Words (for example, 'plus', 'minus', 'equals').
- Mathematical symbols (for example, [latex]+[/latex], [latex]-[/latex], [latex]=[/latex]).
The same mathematical symbols are used in every language, but you still need to know how to say them in English. Learning to read out symbols will help you understand your teachers, communicate mathematics with others, and read mathematical texts more easily.
Given a function [latex]f(x)[/latex], there are two types of integrals of [latex]f(x)[/latex] that we can define: the indefinite integral and the definite integral.
Firstly, we denote the indefinite integral of [latex]f(x)[/latex] by
\[\int f(x) \,dx.\]
This is read out as the integral of f of x d x. Note that this can also be read out as the integral of f of x with respect to x.
We call it an indefinite integral since we do not specify the lower and upper limits of integration. In the notation of the indefinite integral [latex]\int f(x)\, dx[/latex], [latex]f(x)[/latex] is called the integrand, and the dummy variable [latex]x[/latex] is called the variable of integration. The symbol [latex]\int[/latex] is called an integral sign.
Language Focus
Observation 1
In the paragraph above, the phrase dummy variable is used to indicate that the variable is only a placeholder in the calculation of the integral and does not appear in the final result. Note that, we could replace the dummy variable [latex]x[/latex] with any other letter. For example, [latex]\int f(t) \,dt[/latex] has exactly the same meaning.
Observation 2
Again in the paragraph above, we have the word integrand which is clearly related to the terms integral and integration. The term integrand is composed of two elements integr- and the ending -and. Its meaning is 'a function that is to be integrated'.
What these analogous terms might mean: multiplicand, summand, addend? The final one means 'a number that is to be added'. For example, in the equation [latex]5x + 3 = 18[/latex], the terms [latex]5x[/latex] and [latex]3[/latex] are the addends.
A wider learning point here is that paying attention to the elements that make up words, including technical terms, can help with the understanding of other words that contain those elements. Consider, for example, the term polygon which means a many-cornered shape or a shape with many corners. The element -gon comes from a Greek word meaning corner or angle. Knowing this may help us with the terms triangle, pentagon, hexagon, diagonal, orthogonal and trigonometry. These mean, respectively, three-angled shape, five-angled shape, six-angled shape, through the corner(s), at a right angle and measurement of three-angled shapes.
Many STEM terms come from Greek. Understanding these roots helps you learn related words together.
| Greek root | Meaning | Example terms |
|---|---|---|
| -gon | relating to angles/corners/shapes | pentagon, hexagon, diagonal |
| -metry | the measurement of... | geometry, trigonometry |
| -scope | an instrument for viewing... | microscope, telescope |
| -graph | a visual representation / to write, record | chronograph, bibliography |
| poly- | many | polygon, polynomial |
| -ology | study of | biology, geology |
If [latex]\frac {d} {dx} F(x) = f(x)[/latex], then [latex]\int f(x) \, dx = F(x) + C[/latex]. The function [latex]F(x)[/latex] is called the antiderivative of [latex]f(x)[/latex], and [latex]C[/latex] is called the constant of integration. From this result we can see that integration is the reverse operation of differentiation.
Secondly, we denote the definite integral of [latex]f(x)[/latex] from [latex]a[/latex] to [latex]b[/latex] by
\[\int _a ^b f(x) \, dx.\]
This is read out as the integral from a to b of f of x dx.
In the notation of definite integral [latex]\int _a ^b f(x) \, dx[/latex], [latex]a[/latex] and [latex]b[/latex] are called the limits of integration, where [latex]a[/latex] is the lower limit and [latex]b[/latex] is the upper limit. We call this the definite integral, since we specify the limits of integration, as opposed to the indefinite integral above.
Mathematics Focus
Note that the definite integral is a constant, while the indefinite integral is a family of functions whose derivate is the integrand [latex]f(x)[/latex].
Furthermore, the integral sign [latex]\int[/latex] is an elongated S, for summation, reminding us that integration is fundamentally about summing the areas of lots of little strips, of height [latex]f(x)[/latex] and width [latex]dx[/latex].
The Fundamental Theorem of Calculus allows us to evaluate the definite integrals of functions, according to
\[\int_a^b f(x)\,dx = F(x)\bigg|_a^b.\]
The right-hand side is read out as F of x is evaluated at a and b. The right-hand side is just notation that means [latex]F(x)\bigg|_a^b = F(b)-F(a).[/latex]
The process of calculating an integral is called integration and the verb is integrate. We can also say to take the definite/indefinite integral of a function.
Language Focus
Be careful not to confuse the mathematical meanings of the words integration and integrate with their general meanings. The general meanings of the words integration and integrate mean that two things become closely related or combined together to form a whole. For example, we can say: Writing about a topic helps you integrate new knowledge with what you already know.
There are many examples of mathematical terms with both technical and general meanings. Some examples are function, power, term, series and derivative. You may wish to keep a list of such mathematical terms which also have a general, non-technical meaning.
A common technique in integration is partial fraction decomposition, which is used to integrate rational functions. It allows us to decompose (break down) a rational function into the sum of simpler rational functions that are easier to integrate. An example is
\[\frac{1}{x^2-x-2}=\frac{1}{(x-2)(x+1)}=\frac{1/3}{x-2}-\frac{1/3}{x+1} .\]
The resulting fractions [latex]\frac{1/3}{x-2}[/latex] and [latex]\frac{1/3}{x+1}[/latex] are called partial fractions.
In the next section, you will learn how to combine mathematical symbols to make complete mathematical statements, instead of studying symbols separately as you did in this section.
