Definite integrals and areas Cambridge IGCSE Additional Mathematics revision
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In plain words
An integral with limits gives a number, and that number is an area: the area between the curve and the x-axis. With a little care it also gives the area trapped between a line and a curve, or between two curves.
6 things to know
- A definite integral has limits. Integrate, then work out the value at the upper limit minus the value at the lower limit. No "+ c" is needed.
- The area between a curve and the x-axis, from x = a to x = b, is the definite integral of y between those limits, as long as the curve is above the axis.
- An area below the x-axis comes out negative. Give the area as a positive number, and if the curve crosses the axis, find the two parts separately.
- The area between two graphs is the integral of (upper − lower) between the points where they meet.
- So first find where they meet, by putting the two equations equal.
- The area under a straight line can often be found more quickly as a triangle or a trapezium.
Worked example
Evaluate the integral of 3x² + 2 between x = 1 and x = 3.
- Integrate: x³ + 2x.
- At x = 3: 27 + 6 = 33. At x = 1: 1 + 2 = 3.
- 33 − 3 = 30.
Worked example
Find the area enclosed between the curve y = x² and the line y = x + 2.
- They meet where x² = x + 2: x² − x − 2 = 0, so x = −1 or x = 2.
- Between these, the line is above the curve. Integrate (x + 2 − x²): x²/2 + 2x − x³/3.
- At x = 2: 2 + 4 − 8/3 = 10/3. At x = −1: 1/2 − 2 + 1/3 = −7/6.
- Area = 10/3 − (−7/6) = 27/6 = 4.5.
Tips and tricks
- Show the integrated expression in square brackets with the limits, then both substitutions. Each is worth a mark.
- Subtract the right way round: upper limit minus lower limit, and upper graph minus lower graph.
It lands in your notebook with its questions as flashcards.
Definite integrals and areas: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
What is the value of the integral of 2x between x = 0 and x = 3?
[x²] from 0 to 3.
What does a definite integral of a curve above the x-axis represent?
Between the two limits.
To find the area between a line and a curve, what must be found first?
Their x-coordinates are the limits.
A definite integral for a region below the x-axis comes to −5. What is the area?
An area is positive.
What is the value of the integral of 3x² between x = 1 and x = 2?
[x³] from 1 to 2: 8 − 1.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
3 questions, 12 marks. Write your answers on paper, then check them.
Definite integrals and areas
Cambridge IGCSE Additional Mathematics 0606 · 12 marks · papermunch.org
Name ______________________________ Date ______________
Evaluate the integral of 4x − x² between x = 0 and x = 2.[3]
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16/3. [2x² − x³/3] gives 8 − 8/3.
Find the area enclosed by the curve y = 6x − x² and the x-axis.[4]
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36. The curve meets the axis at x = 0 and x = 6, and [3x² − x³/3] gives 108 − 72.
Find the area enclosed between the curve y = x² and the line y = 4.[5]
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32/3. They meet at x = −2 and x = 2. The integral of 4 − x² is 4x − x³/3, which gives 16/3 − (−16/3).
Answers: Definite integrals and areas
- 1. 16/3. [2x² − x³/3] gives 8 − 8/3.
- 2. 36. The curve meets the axis at x = 0 and x = 6, and [3x² − x³/3] gives 108 − 72.
- 3. 32/3. They meet at x = −2 and x = 2. The integral of 4 − x² is 4x − x³/3, which gives 16/3 − (−16/3).