Real Oxford and Cambridge integration and curve sketching questions with full model answers, written by Oxford & Cambridge academics.
Book a Free ConsultationIntegration and curve sketching are the two most frequently combined mathematical skills in Oxford and Cambridge Maths, Further Mathematics, and Physics interviews. They are combined because they are deeply connected: a correct sketch of a curve often reveals directly how to set up an integral, and an integral evaluated incorrectly often exposes errors that a sketch would have caught. Our integration and curve sketching pack contains a comprehensive range of interview-style questions covering both skills individually and their powerful combination, each with a full step-by-step model answer written by Oxford and Cambridge academics.
Download free sample ↓ View all packs and purchase →The combination of integration and curve sketching in a single question is not arbitrary. At undergraduate level — and specifically in the first year of a Mathematics or Physics degree at Oxford or Cambridge — students routinely need to sketch a function to understand its behaviour before integrating it to find areas, volumes, or physical quantities. The interview tests whether you have already developed this habit of mind: the instinct to draw before you compute.
When asked to find the area enclosed between two curves, a student who sketches both curves first will immediately see which is on top, where they intersect (giving the limits of integration), and whether the enclosed region is above or below the x-axis. A student who goes directly to algebra risks integrating in the wrong order, using incorrect limits, or obtaining a negative area and not noticing the error. The sketch is not decoration; it is an essential step in the integration problem.
Conversely, interviewers sometimes present an integral and ask the candidate to interpret it geometrically — to draw the region whose area the integral represents, or to reason about the integral's sign and approximate magnitude from a sketch rather than computing it. This direction — from integral to sketch — tests whether you understand integration as a geometric concept rather than just a computational procedure. Both directions are tested in Oxford and Cambridge interviews, and our pack contains questions of both types.
A-level integration covers substitution, by-parts, and standard forms. Oxbridge interviews go further — not in the sense that they require knowledge of university-level techniques, but in the sense that they test flexible and creative application of A-level techniques to unfamiliar integrals. The key integration methods tested are as follows.
Integration by parts is tested repeatedly, often in cascading form: ∫xⁿeˣdx requires integration by parts applied n times, and the student who understands why (because each application reduces the power of x by one) will be faster and more confident than one who has only practised single applications. Interviewers also test the case where integration by parts loops: ∫eˣsin(x)dx requires integration by parts twice to return to the original integral, which can then be solved algebraically as an equation. Recognising when this loop occurs — and knowing to solve for the original integral rather than continuing to apply parts — is a significant differentiator.
Trigonometric substitution appears in integrals involving √(a²−x²), √(x²+a²), and √(x²−a²). For √(a²−x²), the substitution x = a sin θ transforms the integral into one involving cos θ, which is typically straightforward. For √(x²+a²), use x = a tan θ; for √(x²−a²), use x = a sec θ. The student who knows which substitution to apply in each case, and can explain why (because the relevant trigonometric identity eliminates the square root), demonstrates real command of integration technique.
Partial fractions are required for integrands that are rational functions with factorisable denominators. For a denominator with two distinct linear factors, the decomposition is standard. For repeated factors, a repeated fraction term is needed. For irreducible quadratic factors, a linear numerator is required. Interviewers sometimes present integrals where the degree of the numerator is equal to or greater than the degree of the denominator — which requires polynomial long division before partial fractions can be applied. Getting this step right unprompted is a significant marker of mathematical sophistication.
| Integration Technique | When to Apply | Common Pitfall |
|---|---|---|
| Integration by parts | Products of polynomials with exp/trig/log | Forgetting to differentiate u or integrate dv |
| Substitution (algebraic) | Composite functions; u simplifies the integral | Forgetting to substitute dx or transform limits |
| Trig substitution | Integrals with √(a²±x²) or √(x²−a²) | Incorrect choice of substitution; forgetting to back-substitute |
| Partial fractions | Rational functions with factorisable denominators | Not doing long division first if degree(num) ≥ degree(denom) |
| Recognition of standard forms | ∫1/(x²+a²)dx = (1/a)arctan(x/a)+c | Confusing with ∫1/√(a²−x²)dx = arcsin(x/a)+c |
| Completing the square | Quadratic in denominator that won't factorise | Sign errors in completing the square step |
Preparing for integration and curve sketching questions in your Oxbridge interview?
Our integration and curve sketching pack covers the full range of question types — by parts, substitution, partial fractions, area calculations, improper integrals, sketch-to-integral, and physically contextualised integration — each with full model answers. Written by Oxford & Cambridge academics.
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View all packs and purchase →The curve sketching approach described in detail in our graph sketching pack applies equally here — but with a specific focus on how the sketch informs integration. When setting up an area calculation or volume of revolution, the sketch is not optional: it determines the limits, the sign convention, and often the most efficient integration strategy.
When asked to find the area between two curves y = f(x) and y = g(x), the first step is always to sketch both curves on the same axes and identify their intersection points. The intersections give the limits of integration; the sketch shows which function is above the other in the relevant interval; and the area is then ∫[f(x)−g(x)]dx (or ∫[g(x)−f(x)]dx if g is on top). Without the sketch, candidates frequently integrate in the wrong order or use incorrect limits — and in an interview setting, the error is immediately visible to the interviewer.
Volume of revolution questions — find the volume generated when the region bounded by y = f(x) and the x-axis is rotated 360° about the x-axis — require sketching the region, identifying the limits, and applying the formula V = π∫y²dx. A common error is to apply the formula to a region where the curve is below the x-axis, generating a negative value for y² that the formula then squares to give a positive result for the wrong region. The sketch prevents this error by making the relevant region visually explicit before the integration begins.
Improper integrals — integrals with infinite limits or integrands with singularities within the interval of integration — appear in Oxford and Cambridge interviews as a test of whether candidates understand integration as a limit process rather than just a mechanical computation. The most important improper integrals are: ∫₁^∞ (1/xᵖ)dx (convergent for p > 1, divergent for p ≤ 1), ∫₀^1 (1/xᵖ)dx (convergent for p < 1, divergent for p ≥ 1), and ∫₋∞^∞ e^(−x²)dx (the Gaussian integral, equal to √π, proved using the polar coordinates trick).
The approach to any improper integral is the same: replace the problematic limit with a parameter (call it t or R), evaluate the definite integral with that parameter, and then take the limit as the parameter approaches the problematic value. If the limit exists and is finite, the integral converges; if not, it diverges. Stating this procedure explicitly — and narrating it as you apply it — demonstrates understanding of the foundational definition of integration as a limit rather than just a computational formula.
Interviewers sometimes ask for the physical interpretation of a convergent improper integral: what does it mean that ∫₁^∞ (1/x²)dx = 1? This is asking you to interpret the area under the curve y = 1/x² for x ≥ 1 as a finite number, which requires reconciling the intuition that an infinitely long strip must have infinite area with the mathematical fact that the decay of the function is fast enough to keep the total area bounded. Reasoning about this clearly — and connecting it to the comparison test for series — is the kind of discussion Oxford and Cambridge interview rooms are built for.
First-order differential equations with separable variables appear in Oxford and Cambridge Maths and Physics interviews as integration problems in disguise. A separable ODE has the form dy/dx = f(x)g(y), which can be rearranged as dy/g(y) = f(x)dx and integrated on both sides. The most common examples are: exponential growth/decay (dy/dx = ky, with solution y = Ae^(kx)), Newton's law of cooling (dT/dt = −k(T − T₀)), and simple harmonic motion (d²x/dt² = −ω²x).
What interviewers expect beyond simple separation of variables is the ability to apply initial conditions correctly, check the physical dimensions of the solution, and interpret the result. 'Does your solution make physical sense as t → ∞?' and 'what is the long-term behaviour of the temperature in Newton's law of cooling?' are standard follow-up questions. The curve sketching connection appears here too: once you have derived the solution, you should be able to sketch it and describe its qualitative behaviour — which is more informative for physical understanding than the formula alone.
Sketch the curve y = x·e^(−x) for x ≥ 0. Find the area between the curve and the x-axis for x ∈ [0, ∞). Without further calculation, state whether the area between the curve y = x²·e^(−x) and the x-axis over the same interval is larger or smaller, and justify your answer from the sketch.
The model answer sketches y = xe^(−x): note y(0) = 0, y → 0 as x → ∞, stationary point at x = 1 (where dy/dx = e^(−x) − xe^(−x) = e^(−x)(1−x) = 0), so maximum at (1, 1/e). The area is ∫₀^∞ xe^(−x)dx, evaluated by integration by parts: let u = x, dv = e^(−x)dx, giving [−xe^(−x)]₀^∞ + ∫₀^∞ e^(−x)dx = 0 + [−e^(−x)]₀^∞ = 1. For the comparison: x²e^(−x) ≤ xe^(−x) for x ∈ [0,1] (since x² ≤ x) and x²e^(−x) ≥ xe^(−x) for x ≥ 1 (since x² ≥ x) — so neither is uniformly above the other. The comparison of areas requires evaluating ∫₀^∞ x²e^(−x)dx = 2 by integration by parts applied twice. The model answer explains both the calculation and the geometric reasoning. Download the free sample for more worked examples.
"I had no idea what to expect from my Maths interview at Magdalen — A-level gives you no preparation for the style of question they ask. Working through the pack beforehand meant I'd practised thinking through problems I'd never seen before. When I got stuck in the actual interview, I knew how to keep going rather than freeze. I got my offer in January."— James H., Mathematics, Magdalen College Oxford, 2024 entry
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Oxford and Cambridge Mathematics interviews test flexible application of A-level integration techniques to unfamiliar integrals. The most commonly tested methods are integration by parts (including cascading and looping applications), algebraic and trigonometric substitution, partial fractions (including with repeated or irreducible quadratic factors), completing the square for quadratic denominators, and recognition of standard forms such as ∫1/(x²+a²)dx and ∫1/√(a²−x²)dx. The key skill is identifying which technique is appropriate for an unfamiliar integral and executing it while explaining the rationale.
Interviewers frequently use the sketch to set up the integral and the integral to give quantitative precision to the sketch. For area-between-curves questions, the sketch is necessary to identify which function is on top, find the intersection points (limits of integration), and confirm the sign of the result. Conversely, interviewers may ask you to interpret an integral geometrically — draw the region it represents, or reason about its sign from a sketch — without computing it. Both directions are tested, and our pack covers both with full model answers.
State your substitution explicitly, explain why it simplifies the integral, transform both the integrand and the differential completely before integrating, and for definite integrals, transform the limits as well. Example: for ∫x√(1−x²)dx, say 'I'll substitute u = 1−x², so du = −2x dx, meaning x dx = −du/2. The integral becomes −(1/2)∫√u du = −(1/3)u^(3/2) + c = −(1/3)(1−x²)^(3/2) + c.' This narration demonstrates understanding of what the substitution achieves, not just mechanical execution.
A frequently tested question type is: given the sketch of y = f(x), sketch y = f'(x) without differentiating algebraically. This requires knowing that f'(x) is zero at maxima and minima of f(x), positive where f(x) is increasing, negative where f(x) is decreasing, and has extrema at inflection points of f(x). The reciprocal — given f'(x), recover qualitative features of f(x) by graphical integration — also appears. Both questions test whether you understand calculus as a geometric operation rather than just an algebraic procedure.
Yes — integration is central to Physics and Engineering interviews but in a physically contextualised form. Examples include: computing work done by a variable force (W = ∫F dx), finding electric potential due to a continuous charge distribution, computing the moment of inertia of a rod or disc (I = ∫r²dm), or finding the gravitational potential energy of a system. The approach is the same as in pure mathematics but the setup and interpretation are physical, and the answer should be checked for correct dimensions and plausible magnitude.
Leading Tuition's integration and curve sketching pack covers the full range of question types — by parts, substitution, partial fractions, area calculations, improper integrals, sketch-to-integral, derivative sketching, and physically contextualised integration — each with a full step-by-step model answer showing the logical structure and key insights. Mock sessions with our Oxford and Cambridge Maths and Physics academics replicate the interview format with detailed written feedback. Download the free sample to see the question style, then visit our resources page to explore the full pack. Rated Excellent on Trustpilot (4.8/5).
Further Reading: Integration and curve sketching are core Maths interview skills. For the wider Oxford Maths interview format and additional worked examples, see our companion guide: Oxford Maths Interview Questions 2026 — Step-by-Step Model Answers.
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See also: Maths Oxbridge Interview Questions, Graph Sketching Oxbridge Interview Questions, Derivations Oxbridge Interview Questions, and MAT preparation.
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