IB Maths: Integration & Area Under Curves — FAQ

Read the question for what it's actually asking, then sketch the curve. Find where it crosses the x-axis or the other curve, split the integral at each crossing, and take the absolute value of any negative regions before evaluating using ${\int }_a^{b}f(x)dx=F(b)-F(a)$.

An indefinite integral gives a family of functions plus a constant of integration, C — it reverses differentiation without fixed limits. A definite integral, written ${\int }_a^{b}f(x)dx$, has limits a and b and evaluates to an actual number: the signed area between the curve and the x-axis over that interval.

Solve f(x) = g(x) to find the intersection points — these become your limits. Then integrate upper minus lower function between them: ${\int }_a^b[f(x)-g(x)]dx$. If the curves swap which one's on top partway through, split the integral at that crossing and add the two areas as positive values.

A negative definite integral just means the curve sits below the x-axis over that interval — the integral gives signed area, not actual area. If a question asks for "the area enclosed" or "the region bounded", take the absolute value, or split the integral at any root and add the magnitudes together.

Both SL and HL integrate polynomials, trig functions and $e^x$, and find areas and basic volumes of revolution. HL Maths (AA or AI) goes further — substitution with more complex functions, integration by parts, and using integration to solve differential equations, none of which appear on the SL syllabus.

Yes — substitution appears on both AA SL and AI SL, but only for reasonably direct cases where the substitution is given or easy to spot. According to the IB, the current Analysis & Approaches and Applications & Interpretation guides were first examined in 2021, and substitution sits firmly under SL Topic 5, calculus.

Yes — integration by parts is HL-only in both AA and AI, used for products like $\int x{e}^{x}dx$. SL students never need it. If you spot it on a paper you're using for practice, double-check it's actually an SL paper and not a mislabelled HL one.

The same handful of errors show up every session: forgetting to split the integral at a sign change, integrating upper-minus-lower the wrong way round, dropping the +C on an indefinite integral, or misreading limits off a sketch. Most lost marks are setup errors, not integration errors.

Calculus — differentiation and integration together — makes up roughly 30% of the AA syllabus and around 20-25% of AI, spread across Paper 1 and Paper 2. Area-under-curve questions specifically appear in almost every session, often worth 6-10 marks inside a longer Paper 2 question.

Yes — in Paper 2, your GDC can evaluate definite integrals numerically, which is often faster and safer than integrating by hand under time pressure. You still need to show analytic method marks where the question asks for it, and Paper 1 requires working entirely without a calculator.

Rotating about the x-axis uses $V=\pi {\int }_a^b[f(x){]}^{2}dx$; rotating about the y-axis means rewriting x in terms of y first, giving $V=\pi {\int }_c^d[g(y){]}^{2}dy$. This is HL-only content in both AA and AI, and it usually appears in a longer Paper 2 or Paper 3 question.

Get the standard integrals — polynomials, trig, exponential, 1/x — automatic, so you're not burning exam time recalling them. Then drill past-paper area and volume questions until splitting integrals at sign changes feels routine. Grade 7 scripts rarely show wrong methods; they lose marks on sloppy limits or a missing absolute value.

Area between a curve and the x-axis, area between two curves, and volume of revolution (HL) recur almost every session. Kinematics-linked integration is also common — using $\int v(t)dt$ to find displacement from velocity — alongside the trapezoidal rule for approximating integrals numerically.

Topical worksheets that isolate area-under-curve and volume-of-revolution questions work better than working straight through a textbook chapter, because the skill is repetitive — spotting sign changes, setting correct limits — and needs targeted reps. On RevisionPrep, Topical Worksheets and Mock Papers group these questions by exact sub-skill for that reason.

No — most students can master integration through structured self-study: worked Revision Notes, a question bank for repeated practice, and full Mock Papers done under timed conditions. The skill is procedural, so consistent, feedback-driven practice usually matters more than one-on-one teaching hours for this specific topic.