3.4 Integration: Equations and Diagrams
In the previous section, you learnt how to say some mathematical symbols used in integration. Now, you will practise reading out full equations and labelling diagrams. This will help you use mathematical language in real examples. By practising these skills, you will become more confident when speaking about mathematics in lectures, explaining your work to others, and understanding what your lecturers and classmates say during lectures, seminars, and tutorials.
In this first activity, you will practise reading out some equations. This will help you improve your pronunciation and become more confident when talking about mathematics.
For each equation, first take a moment to consider how you would say it, then try reading it out loud. After reading each equation aloud, select the 'Answer' button below the equation to see the correct phrasing, and use the 'Play' button to listen to the recording.
You can also do this activity with a partner or in a small group. Working together lets you take turns, help each other, and share what you know about reading mathematics in English.
How would you read out the equations below?
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- [latex]\int _a ^b f(x) \, dx = F(x) \bigg| _a ^b=F(b)-F(a)[/latex]
Answer
The integral from a to b of f of x d x is equal to, capital f of x evaluated at a and b, which is equal to, capital f of b minus, capital f of a.
- [latex]\int _a ^b f(x) \, dx = F(x) \bigg| _a ^b=F(b)-F(a)[/latex]
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- [latex]\int x^n \, dx = \frac {1} {n+1} x^{n+1} + C[/latex]
Answer
The integral of x to the power of n d x is equal to, one over n plus one, times, x to the power of n plus 1, plus C.
- [latex]\int x^n \, dx = \frac {1} {n+1} x^{n+1} + C[/latex]
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- [latex]\int \sin x \, dx = -\cos(x) + C[/latex]
Answer
The integral of sine x d x is equal to, minus cos x + C.
- [latex]\int \sin x \, dx = -\cos(x) + C[/latex]
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- [latex]\int \tan x \, dx = -\ln |\cos x| + C[/latex]
Answer
The integral of tan x d x is equal to, minus l n of the absolute value of cos of x, plus C.
- [latex]\int \tan x \, dx = -\ln |\cos x| + C[/latex]
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- [latex]\int \frac {8}{(x-1)(x^2+2x+5)} \, dx =\ln|x-1| -\frac{1}{2}\,\ln(x^2+2x+5)-\tan^{-1}\Bigl(\frac{x+1}{2}\Bigr)+C[/latex]
Answer
The integral of eight over, x minus 1 times, x squared plus two x plus five, d x, is equal to, l n of the absolute value of x minus one, minus a half l n of x squared plus two x plus 5, minus arctan of, x plus 1 over 2, plus C.
- [latex]\int \frac {8}{(x-1)(x^2+2x+5)} \, dx =\ln|x-1| -\frac{1}{2}\,\ln(x^2+2x+5)-\tan^{-1}\Bigl(\frac{x+1}{2}\Bigr)+C[/latex]
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- [latex]\int \frac {\sqrt {x^2-1}} {x} \, dx =\sqrt{x^2-1}-\sec^{-1}(x)+C[/latex]
Answer
The integral of the square root of x squared minus one all over x dx, is equal to, the square root of x squared minus one, minus arcsec of x, plus C.
- [latex]\int \frac {\sqrt {x^2-1}} {x} \, dx =\sqrt{x^2-1}-\sec^{-1}(x)+C[/latex]
In this second activity, you will do the opposite of the previous activity. Before, you saw a mathematical equation and said it in English. Now, you will read and listen to the English 'read-out' by pressing the 'Audio' button, and then write down the correct mathematical equation. After writing down the equation, select the ‘Turn’ button to reveal the correct answer, then select the ‘Next’ button to proceed.
For example, if you hear or read A squared plus B squared is equal to C squared, you should write [latex]A^2+B^2=C^2[/latex].
This activity will help you practise turning English sentences into mathematical symbols. It will also improve your listening, reading, and writing skills in mathematics, making it easier to understand and communicate mathematical ideas.
In this final activity, you will continue to practise using both mathematical language and objects together. You will be asked to label certain parts of the diagrams using the correct technical terms.
This activity will help you connect mathematical vocabulary to real examples, such as graphs, shapes, or other visuals. By labelling diagrams, you will learn how to describe mathematical objects more clearly and accurately. This skill is useful when explaining your ideas, answering questions in class, or understanding diagrams in textbooks and exams.
Up to now, you have focused on reading and speaking mathematical language through equations and diagrams. You have learned how to say individual symbols and how to read complete expressions aloud. In the next section, you will shift focus to mathematical communication between people, to learn how mathematicians and students use this language when discussing problems, asking questions, and explaining ideas to each other.
