TMUA Integration: A Topic Guide

Integration is the TMUA maths topic listed in the syllabus as MM7, covering indefinite and definite integrals of powers of x, the relationship between a definite integral and the area between a curve and the axes, the fundamental theorem of calculus, combining and splitting integrals, the trapezium rule, and differential equations whose derivative is already written as a function of x alone. In the FrontierVUE catalogue of TMUA questions, 98 are tagged to this topic; 38 of them come from past papers, with every year from 2016 to 2023 contributing some, so within that catalogue it is regular rather than occasional content. Its difficulty sits in the middle band: 59 questions are rated MEDIUM, 27 EASY, 11 HARD and 1 VERY_EASY. Integration is therefore a bank of marks you are expected to secure rather than a top-end differentiator.

TMUA integration at a glance

Syllabus code: MM7, with 6 syllabus points (MM7.1 to MM7.6)
Questions catalogued: 98 questions tagged to this topic in the FrontierVUE catalogue
Difficulty mix: 59 MEDIUM, 27 EASY, 11 HARD, 1 VERY_EASY, as catalogued by FrontierVUE
Years covered: In the FrontierVUE catalogue the topic appears in every year from 2016 to 2023; 2021 and 2023 each carry 7, the highest counts
The two heaviest points: MM7.2 integrating powers, 62 questions; MM7.1 integrals and area, 38, in the FrontierVUE catalogue
Use in mock papers: 33 placements of these questions on mock papers in the FrontierVUE catalogue

Question counts, difficulty ratings and year coverage describe the FrontierVUE catalogue, not an official breakdown of the papers; the exam board does not publish per-question topic tags. A single question may carry more than one syllabus tag, so the per-point counts are not additive. The syllabus descriptions paraphrase the official test information page.

What integration covers on the TMUA

The TMUA syllabus files integration under MM7, which contains six points. They are not six equal blocks: one is the spine, and the rest are the contexts you have to read that spine into, namely area, intervals, approximation and differential equations. Seeing that structure settles the revision order for you: drill the spine until it is automatic, then pick off the contexts one by one.

MM7 Integration

6 syllabus points, 98 questions in the FrontierVUE catalogue

MM7.2 Definite and indefinite integrals of powers62 questions, the largest single point in the FrontierVUE catalogue

Integrating rational powers of x, with the single exponent the power rule cannot handle excluded
Sums and differences handled term by term
Expressions that must be expanded or simplified before they can be integrated term by term

MM7.1 Definite integrals and area38 questions in the FrontierVUE catalogue

The value of a definite integral is not the same object as the area between the curve and the axes
Knowing when the two agree and when the region has to be split

MM7.3 The fundamental theorem of calculus10 questions in the FrontierVUE catalogue

Evaluating a definite integral from an antiderivative at the two endpoints
Differentiating an integral with a variable upper limit to recover the integrand

MM7.4 Combining and splitting integrals10 questions in the FrontierVUE catalogue

Integrals over the same interval combine into one integral of a combined integrand
Integrals over adjacent intervals join end to end

MM7.5 The trapezium rule7 questions in the FrontierVUE catalogue

Approximating the area under a curve with trapezia
Deciding whether the approximation over-estimates or under-estimates

MM7.6 Directly integrable differential equations4 questions in the FrontierVUE catalogue

The derivative is already a function of the variable alone, so a single integration recovers the solution

Tap a branch to unfold

The centre of gravity is unmistakable. In the FrontierVUE catalogue, MM7.2 carries 62 questions and MM7.1 carries 38, while the other four points are far smaller. Note that one question can hold several syllabus tags, so these per-point counts are not meant to be added together. Small counts are still no reason to skip those four outright: their shapes are fixed, so they are quick to close.

How often integration shows up in the catalogue

Questions tagged to this topic in the FrontierVUE catalogue

Syllabus points under this topic

2016 to 2023

Every year in this span carries the topic in the FrontierVUE catalogue

Placements on mock papers in the FrontierVUE catalogue

	Questions in the FrontierVUE catalogue	Where the year sits
2016	5	Steady presence
2017	5	Steady presence
2018	2	The lightest year in the catalogue
2019	5	Steady presence
2020	4	Steady presence
2021	7	Joint heaviest year
2022	3	A lighter year
2023	7	Joint heaviest year

The table counts questions tagged to this topic in the FrontierVUE catalogue; it is not an official breakdown of the papers, since the exam board does not publish per-question topic tags.

The point of the table is not any single year's count but the fact that the column never goes empty between 2016 and 2023. Integration is not a module that surfaces every few years: it turns up in every year covered by the FrontierVUE catalogue, and betting against it has nothing to stand on. Within that catalogue, 2021 and 2023 each carry 7 questions, the two heaviest years, and even the lightest year, 2018, still contains the topic. One further question in that catalogue is tagged to the official specimen papers, which are not one of the sitting years.

Where integration actually gets hard

Of the 98 questions in the FrontierVUE catalogue, 59 are rated MEDIUM, 27 EASY, 11 HARD and 1 VERY_EASY. With the weight in the middle band and only a small hard tail, this reads as a topic to secure rather than one that separates the strongest candidates. In our teaching that shapes what we ask students to check first: dropped terms in simplification, flipped signs and concepts applied backwards, rather than going straight for the 11 hard items.

	Questions in the FrontierVUE catalogue	What it means for revision
MEDIUM	59	The main band: work it until it is both reliable and quick, with expand or reduce first, then integrate term by term as the focus
EASY	27	Marks on offer: one slip here is a pure loss, so leave time to check signs and the constant on indefinite integrals
HARD	11	A small hard tail: save it for late revision and start with combinations such as area judgement plus splitting the region
VERY_EASY	1	Nothing to plan around

Difficulty labels are FrontierVUE catalogue ratings, not an official assessment of the questions.

Reading a definite integral straight off as an area

Where the curve sits below the axis, that stretch contributes negatively to the integral, while an area is always positive. If a question asks for area, locate the crossings first, split the region and take positive values; if it asks for the integral, evaluate it as it stands. MM7.1 asks for an understanding of the difference between evaluating a definite integral and finding the area between a curve and the axes, so the two have to be handled apart.

Integrating first and simplifying afterwards

MM7.2 explicitly includes expressions that must be expanded or reduced before they can be integrated. When a bracketed power, a quotient or a surd form appears, rewrite it into something the power rule can take term by term, and only then integrate. Doing it the other way round tends to produce a form with no clean finish, which you then force through under time pressure.

Forgetting the one exponent the power rule cannot take

The power rule handles rational powers of x in general, with exactly one exponent excluded, and MM7.2 states that exclusion. A common failure under exam conditions is mechanical: the formula gets applied anyway, produces a zero denominator, and the work carries on regardless. Glancing at the exponent before you start costs a second and is worth it.

Computing the trapezium rule without judging it

MM7.5 asks for more than a numerical estimate: you also have to say whether it over-estimates or under-estimates the true area, which depends on how the curve bends across the interval. Stopping at the number is doing half the task, since MM7.5 lists judging over- or under-estimation alongside computing the approximation. The point carries few questions in the FrontierVUE catalogue, but its shape is fixed, so one dedicated pass covers it.

If you drill only two things, drill simplify before you integrate, and decide whether the question wants an integral or an area before you write anything. The two heaviest points in the FrontierVUE catalogue, MM7.2 and MM7.1, map onto exactly those two habits: make them automatic first, then widen out to the other four points.

How to close the gap on integration

On a topic where the middle band dominates, the real risk is feeling fine about it: everything looks familiar on the page and marks disappear as soon as you write. So the route below does not start with notes. It starts with a small set of questions that exposes how you actually go wrong, then works through the syllabus points in order of weight, so you can stop at any stage and still know what you gained.

01Locate

Diagnose first: topic practice, integration only

In topic practice, select this topic alone, work a short set, then separate calculation slips from conceptual errors. Slips are treated with fluency and checking routines; conceptual errors send you back to the syllabus point to re-read what it actually says. The two need entirely different treatment.

02MM7.2

Build the spine: make MM7.2 automatic

Drill the power rule, term by term integration, and the expand or reduce first shapes until they need no thought, while keeping the excluded exponent in view. This is the largest single point in the FrontierVUE catalogue, so starting here covers the most ground.

03MM7.1

Add the context: area questions in MM7.1

Work a set that asks for areas, and open every one with the same question: is the value of the integral the same number as the area here. If you cannot answer, sketch and find the crossings before anything else. This is the second largest point in the FrontierVUE catalogue.

04Sweep

Sweep the rest: MM7.3 through MM7.6

The fundamental theorem, combining and splitting integrals, the trapezium rule and directly integrable differential equations all carry small counts in the FrontierVUE catalogue, but their shapes are fixed. One concentrated session covering all four beats scattering them through your revision.

052016 to 2023

Return to past papers: test by year

Finish in past paper practice, checking whether the topic holds up across the papers from 2016 to 2023. Time pressure in a full paper is the real test: what you get right untimed does not always survive the clock.

This route maps onto topic practice and past paper practice in the FrontierVUE practice centre. This page is open to read; practice and the weakness report require signing in, and the weakness report also requires the relevant subject to be unlocked.

FAQ

Frequently asked questions

Does the TMUA test integration?

Yes. Integration is a named topic in the TMUA maths syllabus, listed as MM7 with 6 syllabus points beneath it. In the FrontierVUE catalogue of TMUA questions, 98 are tagged to it; 38 come from past papers, with every year from 2016 to 2023 represented. Treat it as certain content rather than something you might skip.

How hard are TMUA integration questions?

Mostly mid band. Of the 98 questions in the FrontierVUE catalogue, 59 are rated MEDIUM, 27 EASY, 11 HARD and 1 VERY_EASY. With so few hard items, this is a topic to secure rather than one that separates the strongest candidates. The difficulty labels are FrontierVUE catalogue ratings, not an official assessment of the questions.

Which integration skills should I practise for the TMUA?

Cover the 6 syllabus points: definite and indefinite integrals of powers of x including cases that must be simplified first, the relationship between a definite integral and the area between a curve and the axes, the fundamental theorem of calculus, combining and splitting integrals, the trapezium rule together with judging over or under estimation, and differential equations whose derivative is a function of the variable alone. The first two are the largest in the FrontierVUE catalogue at 62 and 38 questions, and your practice should be weighted the same way.

How much revision time should integration get?

Treat it as steady routine practice rather than a special project. It holds 98 questions in the FrontierVUE catalogue, 38 of them from past papers spanning every year from 2016 to 2023, and sits mainly in the mid band, which suits short sets repeated often: work a few at a time and watch whether the same error keeps returning, instead of clearing it once and never looking back. The 11 HARD items can wait for a concentrated late session.

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