CAIE A-Level Mathematics 2.5.3 Trapezium Rule

Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise applying the trapezium rule with the correct interval width and endpoint weighting and judging over- or under-estimation from the curve's position relative to chords.

How this is tested

calculate each required y-value and identify h from the bounds and number of intervals
apply the full h/2[y₀ + yₙ + 2(y₁ + ⋯ + yₙ₋₁)] formula with endpoints counted once
state over- or under-estimate by comparing the trapezium tops with the curve

[Maximum number: 6]

The diagram shows the curves with equations $y=\sqrt[3]{5 x^{2}+7}$ and $y=\frac{27}{2 x+5}$ for $x \geqslant 0$ .
The curves meet at the point (2,3).
Region A is bounded by the curve $y=\sqrt[3]{5 x^{2}+7}$ and the straight lines x=0, x=2 and y=0.
Region B is bounded by the two curves and the straight line x=0.

(a)

Use the trapezium rule with two intervals to find an approximation to the area of region A. Give your answer correct to 3 significant figures.

[ 3 ]

(b)

Deduce an approximation to the area of region B. Give your answer correct to 3 significant figures.

[ 1 ]

(c)

State, with a reason, whether your answer to part (c) is an over-estimate or an under-estimate of the area of region B.

[ 2 ]

How this is tested

Question 6

Question 6(a)

Question 6(c)

Question 6(d)