About this chapter
Integration in Pure 2 means definite integrals and areas. The examiner reports show a clear split: the trapezium rule is one of the best-answered topics on the paper, while areas involving fractional powers, regions below the x-axis, or a line and a curve together separate the grades.
This chapter covers evaluating definite integrals, rewriting expressions as powers of x first, areas under a curve, regions below the axis, areas between a curve and a line, unknown limits and constants, the trapezium rule and whether it over- or underestimates, and using one integral to find a related one.
The ten questions
- Q1 (4 marks): the exact value of a definite integral with a 4/√x term, from 1 to 4.
- Q2 (5 marks): a definite integral equal to 40 fixes the constant k.
- Q3 (5 marks): the area between y = 6x − x² and the x-axis.
- Q4 (5 marks): complete a table, apply the trapezium rule, and say whether the estimate is too big or too small.
- Q5 (5 marks): an integral of a cubic from 0 to 4 equals zero, and why that is not the area.
- Q6 (8 marks): where a line meets a parabola, then the exact area between them.
- Q7 (8 marks): an integral with an unknown upper limit gives a cubic in k, its only real root, and a related integral by symmetry.
- Q8 (8 marks): the trapezium rule on y = 2 to the power x, why it overestimates, and a related integral from the same estimate.
- Q9 (9 marks): the maximum point of a cubic, then exact areas on either side of it.
- Q10 (10 marks): the exact area under y = 8√x − 2x, a trapezium rule estimate, and its percentage error.
Key skills tested
Definite integrals. Integrate, then substitute the upper limit and subtract the value at the lower limit. Show both substituted brackets, and there is no + c.
Rewrite first. Split fractions into powers of x. For example, (x − 3)/√x becomes √x − 3/√x.
Area under a curve. Integrate y between the limits when the curve is above the x-axis, finding the limits from the roots.
Regions below the axis. There the integral is negative. Integrate each part separately and add their sizes.
Between a curve and a line. Find where they meet, then integrate the top minus the bottom, or subtract the area under the curve from a trapezium.
Unknown limits and constants. Evaluate in terms of k, solve the equation, and check which roots are allowed.
The trapezium rule. The strip width h is (b − a)/n, and n strips need n + 1 values of y.
Over or under? If the curve bends upwards, the trapezia sit above it and the rule overestimates. More strips improve the estimate.
Related integrals. A constant multiple can be taken outside the integral, and a constant c integrates to c(b − a) between the limits.
Worked example
Question 3 from this chapter. The curve C has equation y = 6x − x². (a) Find where C meets the x-axis. (b) Use integration to find the area of the region bounded by C and the x-axis.
(a) 6x − x² = x(6 − x) = 0, so x = 0 and x = 6.
(b) Integrating 6x − x² gives 3x² − x³/3. At x = 6 this is 108 − 72 = 36, and at x = 0 it is 0. The area is 36 − 0 = 36.
The curve is above the x-axis between its roots, so the integral gives the area directly.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- June 2023, Q1: the trapezium rule from a table of values, to a stated accuracy, as in Q4.
- June 2024, Q6: the trapezium rule on an exponential curve, then related integrals worked out from that estimate, as in Q8.
- June 2024, Q9(b): regions above and below the x-axis whose integrals cancel, as in Q5.
- June 2023, Q10: integrating fractional powers, then the area between a curve and a straight line, as in Q2 and Q6.
Where marks are lost
- The strip width. In June 2023 a surprising number could not find h, even though the table showed consecutive values of x, and it was still the usual error in June 2024.
- Brackets in the trapezium rule. The inner and outer brackets were the main source of wrong answers in June 2023.
- Subtracting the limits. In June 2023 many answers went wrong at the subtraction, often through a missing bracket around the lower-limit value.
- Starting again instead of using an earlier part. In June 2024, applying the trapezium rule a second time instead of using the first estimate earned nothing.
Common questions
Is the trapezium rule in the formula booklet?
Yes, but you still need to find h correctly and use n + 1 values of y for n strips.
Can an area be negative?
No. If part of a region is below the x-axis, integrate that part separately and use its size.
How do I find the area between a curve and a line?
Find where they meet, then integrate the upper function minus the lower one between those x values.
Where this chapter leads
- Pure 3 Chapter 7: Integration: standard integrals, the reverse chain rule and trigonometric integrals.
- Pure 4 Chapter 6: Integration: substitution, integration by parts and volumes of revolution.