A Level · Cambridge (CIE) · 9709

Choose exact integration or data estimates and explain their limits

Connect methods to the evidence available, keep signed and accumulated quantities distinct and avoid treating agreement or refinement as proof of accuracy.

Section 2.5 · Integration

Pure Mathematics 2 · Integration · 2.5

Exam scope and assessment notes

Pure Mathematics 2 (Paper 2, AS only) · Mathematics 9709, 2026–2027. Paper 1 integration and Paper 2 derivatives and identities are assumed. The AS route using Papers 1 + 2 cannot be carried forward to A Level. Section 2.5 covers reverse differentiation of linear-composite exponentials, reciprocals and trigonometric functions, integration using trigonometric relationships, and trapezium estimates with sketch-based over/under-estimation. The general method of integration by substitution is not required here. Use radians for trigonometric calculus. Written practice is self-checked.

Before you begin

Integrate the section's linear natural and trigonometric forms and use identities.
Calculate trapezium estimates and judge chord positions using curve sketches.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

What information permits an exact actual integral to be inferred from a finite data table alone?

What you will learn

Choose a suitable exact primitive or trapezium estimate for a supplied function or measurement table.
Interpret units, initial values, sign changes and what the calculation does or does not establish.

Use an exact primitive when the function and method permit it

A supplied expression may give a directly checkable exact integral, such as ∫ from 0 to ln 2 of (2eᵗ + 1) dt = [2eᵗ + t] from 0 to ln 2 = 2 + ln 2. If a question requests a trapezium approximation, use its stated partition even when an exact primitive also exists; an exact value can then check the approximation. If only measurements are supplied, the unknown between-point curve generally prevents an exact actual integral from being inferred from the table.

A velocity integral gives signed displacement, while distance requires splitting at velocity sign changes and adding the magnitudes of the separate displacements. For v(t) = cos t cm/s on [0, π] seconds, the displacement is 0 but distance is 2 cm, split at t = π/2. A two-strip signed trapezium estimate also gives 0 from ordinates 1, 0, −1. Its agreement with the displacement does not prove the particle stayed still or that an area-magnitude estimate is exact.

Refinement is useful evidence, not an automatic guarantee

Suppose measured inflow rates at t = 0, 1, 2, 3, 4 minutes are 2, 5, 6, 4, 1 litres per minute. The four-strip estimate is (1/2)[2 + 1 + 2(5 + 6 + 4)] = 16.5 litres. Using only times 0, 2, 4 gives a two-strip estimate 15 litres. The difference 1.5 litres shows sensitivity to the sampled shape; it is not a proven maximum error relative to the actual unknown integral. An unmeasured bump can keep every recorded ordinate the same while changing the integral.

If the initial stock is 20 litres and there is no outflow, the finer estimate gives final total 36.5 litres. Report the interpolation-based estimate, its time interval and units. State any model or sampling limitation relevant to the requested conclusion. Do not label a data estimate as an exact measured total, or claim a target accuracy solely because two successive estimates are close. For power data in kW sampled in minutes, convert widths to hours before reporting kWh.

Pause and explain

For v(t) = cos t on [0, π], does displacement 0 imply distance 0?

Put the idea to work

Worked example

Measured inflow rates at 0, 1, 2, 3, 4 minutes are 2, 5, 6, 4, 1 litres per minute. A vessel starts with 20 litres and has no outflow. Estimate its final volume and compare with the two-strip estimate from times 0, 2, 4.

Show the worked solution

Four width-1 strips give estimated increase (1/2)[2 + 1 + 2(5 + 6 + 4)] = 16.5 litres.
The final total estimate is 20 + 16.5 = 36.5 litres. Two width-2 strips from ordinates 2, 6, 1 give increase 15 litres.
The difference is 1.5 litres; it indicates partition sensitivity but supplies no guaranteed error bound on the unknown actual rate integral.

Answer Estimated increase 16.5 litres; final total 36.5 litres. Coarser increase 15 litres; difference 1.5 litres is not a guaranteed error bound.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

A supplied flow is q(t) = 2eᵗ + 1 litres per minute for 0 ≤ t ≤ ln 2. Find the added volume exactly and explain why an exact method is available.

Give me a hint

The given function has a direct exponential-plus-constant primitive.

Compare my reasoning

A primitive of q is 2eᵗ + t.
Upper-minus-lower evaluation gives (4 + ln 2) − 2 = 2 + ln 2.
The increase is exactly 2 + ln 2 litres within the supplied model; a measured table alone would not provide this same between-point information.

Exact modelled increase 2 + ln 2 litres.

Look for these in your work

Use a verified primitive and both endpoints.
Explain which supplied function information supports an exact modelled integral.

2 · Independent

A particle has supplied velocity cos t cm/s on 0 ≤ t ≤ π seconds. Find its exact displacement and distance, and explain a two-strip trapezium estimate of zero.

Give me a hint

Split the motion at the velocity zero t = π/2 before finding distance.

Compare my reasoning

The signed displacement is [sin t] from 0 to π = 0.
The positive half contributes +1 cm and the negative half −1 cm, giving total distance 2 cm.
Two trapezium strips with ordinates 1, 0, −1 also cancel to zero; that estimates signed displacement and does not mean no motion.

Displacement 0 cm; distance 2 cm; the zero signed trapezium estimate hides cancelling motion.

Look for these in your work

Separate displacement from distance using the velocity sign change.
Interpret the numerical rule as signed accumulation rather than proof of no movement.

3 · Transfer

Measured power at 0, 2, 4, 6 minutes is 1, 3, 2, 4 kW. Estimate energy use in kWh with three trapezia and state whether the measurements prove an exact energy total.

Give me a hint

Convert the common two-minute width to 1/30 hour before weighting the power ordinates.

Compare my reasoning

Three strips have width h = 1/30 hour, with endpoints 1 and 4 and interior ordinates 3 and 2 kW.
The estimate is (1/60)[1 + 4 + 2(3 + 2)] = 15/60 = 1/4 kWh.
Report approximately 0.25 kWh from straight interpolation; the measurements do not establish an exact actual total or a guaranteed accuracy.

Estimated energy 1/4 = 0.25 kWh; the measured table alone does not prove an exact total.

Look for these in your work

Match kW to hour-based widths before accumulating energy.
State the interpolation estimate and its evidence limitation.

Common mistakes

A data table does not determine the true between-point function unless an interpolation model is explicitly given.
Equal signed integrals can hide non-zero distance or area.
Convert time units, add initial stocks separately and avoid unsupported accuracy claims.

Recall without your notes

What can you explain now?

Does a small change between successive trapezium estimates prove a specified accuracy for an unknown measured-rate integral?

Compare with the explanation

No.

The difference only compares those approximations; unobserved between-point behaviour can change the actual integral.

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Make it stick

Choose an exact or numerical method for a function and for a table. Explain signed cancellation, unit conversion, initial stock and why estimate refinement alone does not certify an accuracy.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.