H2 Maths Definite Integrals Formula Sheet | Area & Volume

Q: What does H2 Maths Notes (JC 1-2): 5.4) Definite Integrals cover?
A: Area, volume, and mean-value interpretations for H2 Maths Topic 5.4.

Download: Get the H2 Maths Definite Integrals formula sheet (PDF) for quick revision, or the complete notes (PDF) for the full walkthrough.

Before you revise
Always sketch the region before integrating. Label intercepts, intersection points, and orientation so you do not mix up top/bottom or left/right functions.

• A definite integral is signed area: Sketch before calculating.
• Area, volume, and average value use different setups: Identify the interpretation first.
• Most mistakes come from bounds or orientation: Mark top minus bottom, right minus left, or axis of rotation.

Concrete example: If two curves cross at x = -1 and x = 3, those are your limits. Then decide which curve is above the other before integrating.

Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 5.4 scope is within Section A Pure Mathematics, which is assessed in Paper 1 (100 marks) and Paper 2 Section A (40 marks).

Formulas at a glance

Every result the 9758 syllabus expects you to apply, on one screen. MF27 does not provide the area or volume-of-revolution formulas or the definite-integral properties below. The shared booklet does contain arc-length and surface-area results for the syllabuses that require them, which are different from the volume formulas used here.

Definite-integral properties

Area

Volume of revolution

Fundamental Theorem

• If $F'(x) = f(x)$, then $\int_a^b f(x) \space dx = F(b) - F(a)$

Example -- Using an antiderivative

$$\int_1^4 (3x^2 - 2x + 1) , dx = \left[ x^3 - x^2 + x \right]_1^4 = (64 - 16 + 4) - (1 - 1 + 1) = 51.$$

Areas Between Curves

• For vertical slices: $\int_a^b \left( y_\text{top} - y_\text{bottom} \right) \space dx$

Area setup checkpoint

Before writing the integral, decide what the answer represents.

Misconception check: area is not always the same as $\int_a^b f(x) \space dx$. The definite integral is signed area; a physical region uses positive pieces.

Example -- Area enclosed

Find area between $y = x^2$ and $y = 2x + 3$.

1. Intersections solve $x^2 = 2x + 3$ ⇒ $x = -1, 3$.
2. Area $= \int_{-1}^3 \left(2x + 3 - x^2\right) dx = \biggl[ x^2 + 3x - \dfrac{x^3}{3} \biggr]_{-1}^3 = \dfrac{32}{3}$

Volumes of Revolution

• About x-axis: $V = \pi \int_a^b y^2 \space dx$.
• About y-axis: $V = \pi \int_c^d x^2 \space dy$

Volume axis checkpoint

Before calculating volume, decide whether your slice is perpendicular or parallel to the axis of rotation.

Worked check: Rotating $y = \sqrt{x}$, $0 \leq x \leq 4$

Misconception check: an area integral adds strip areas, but a volume-of-revolution integral adds cross-sectional areas. For disk and washer methods, the radius must be squared because each cross-section is circular.

Example -- Volume

Region under $y = \sqrt{x}$ from $x = 0$ to $x = 4$

$$V = \pi \int_0^4 x \space dx = \pi \biggl[ \dfrac{x^2}{2} \biggr]_0^4 = 8\pi.$$

Example -- Shell method about y-axis

Revolve the region bounded by $y = 2 - x$, the x-axis, and $x = 0$ about the y-axis. Using shells:

$$V = 2\pi \int_0^2 x (2 - x) , dx = 2\pi \left[ x^2 - \dfrac{x^3}{3} \right]_0^2 = 2\pi \left( 4 - \dfrac{8}{3} \right) = \dfrac{8\pi}{3}.$$

Mean Value and Average

• Average value of $f(x)$ on $[a, b]$: $\bar{f} = \dfrac{1}{b - a} \int_a^b f(x) \space dx$

Mean-value interpretation checkpoint

Before substituting into the formula, decide what the integral is averaging.

Worked check: if $f(x)=x^2$ on $[0,3]$, the integral gives total accumulation $\int_0^3 x^2 \space dx = 9$

Misconception check: mean value is not the midpoint output $f((a+b)/2)$ unless the function and interval happen to make those values equal.

Example -- Average height

The mean value of $f(x) = \sin x$ on $[0, \pi]$ is

$$\bar{f} = \dfrac{1}{\pi - 0} \int_0^\pi \sin x , dx = \dfrac{1}{\pi} \left[ -\cos x \right]_0^\pi = \dfrac{2}{\pi}.$$

Integration with Modulus or Piecewise Functions

• Identify breakpoints where expression changes sign.
• Split integral into segments where expression simplifies without modulus.

Modulus split checkpoint

Before integrating a modulus expression, find where the inside expression changes sign. Each interval should then have a clear positive or negative version before you integrate.

Worked check: for $\int_{-2}^3 \lvert x-1\rvert \space dx$, the sign changes at $x=1$. On $[-2,1]$

Misconception check: removing the modulus is not a single algebra step. You are choosing a different expression on each side of the sign-change point.

Example -- Modulus

Evaluate $\int_{-2}^3 \lvert x - 1 \rvert dx$.

1. Break at $x = 1$:

$$\int_{-2}^1 (1 - x) dx + \int_1^3 (x - 1) dx = \biggl[ x - \dfrac{x^2}{2} \biggr]_{-2}^1 + \biggl[ \dfrac{x^2}{2} - x \biggr]_1^3 = \dfrac{13}{2}.$$

Improper Definite Integrals

• Treat any integral with an infinite limit or an unbounded integrand as a limit problem. Replace the problematic bound with a parameter, integrate on the safe interval, then take the limit.

Infinite limits

• For $\int_a^\infty f(x) , dx$, evaluate $\lim_{t \to \infty} \int_a^t f(x) \space dx$

Example -- Tail integral

$$\int_1^\infty \dfrac{1}{x^2} \space dx = \lim_{t \to \infty} \left[ -\dfrac{1}{x} \right]_1^t = \lim_{t \to \infty} \left( -\dfrac{1}{t} + 1 \right) = 1.$$

Vertical asymptotes or interior discontinuities

• If $f(x)$ blows up at an endpoint, replace that endpoint with a parameter approaching the troublesome value.
• If the discontinuity lies inside the interval, split the integral at that point and treat each side as an improper integral.

Example -- Endpoint asymptote

$$\int_0^1 \dfrac{1}{\sqrt{x}} \space dx = \lim_{\varepsilon \to 0^+} \int_{\varepsilon}^1 x^{-1/2} \space dx = \lim_{\varepsilon \to 0^+} \left[ 2x^{1/2} \right]_{\varepsilon}^1 = 2.$$

Example -- Interior discontinuity

$$\int_{-1}^2 \dfrac{1}{(x - 1)^2} , dx = \lim_{t \to 1^-} \int_{-1}^t \dfrac{1}{(x - 1)^2} , dx + \lim_{s \to 1^+} \int_s^2 \dfrac{1}{(x - 1)^2} , dx$$

Each limit diverges, so the original integral diverges.

Convergence quick checks

• Benchmark integrals: $\int_1^\infty \dfrac{1}{x^p} , dx$ converges for $p > 1$

Calculator Workflow

• Use GC definite integral function to verify numeric value after completing manual steps.
• When dealing with piecewise functions, evaluate each segment separately on the GC as a check.
• Document the command (e.g. ∫(2x+3-x^2,x,-1,3)) in working if used.

Exam Watch Points

• Always sketch the region and label axes/limits.
• For revolutions, specify axis clearly and include $\pi$ factor.
• State units (square or cubic) when context demands.
• Justify convergence when evaluating improper integrals.

Want weekly guided practice on Definite Integrals? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.

Common exam mistakes

• Forgetting to change limits when substituting: When you apply a substitution such as $u = g(x)$, the limits must be converted to $u$-values before evaluating. Keeping the original $x$-limits with the new integrand is a routine error that costs method marks - always rewrite the bounds in terms of $u$ or revert to $x$ before substituting back.

FAQ

Is there a formula sheet for H2 Maths definite integrals?
Yes - the "Formulas at a glance" section near the top of this page collects the definite-integral properties, area formulas, and volume-of-revolution formulas used in this topic. MF27 does not provide those area or volume formulas. Its applications-of-definite-integrals section instead lists arc length and surface area of revolution about the $x$-axis for the syllabuses that require them.

Which papers examine definite integrals, and how much of the paper do they cover?
Definite integrals (Topic 5.4) fall under Pure Mathematics. For the 9758 syllabus (first exam 2026), Pure Mathematics is examined in Paper 1 (3 hours, 100 marks) and in Paper 2 Section A (approximately 40 marks). [1] Area and volume questions appear in both papers; improper integrals tend to appear in Paper 1. Expect at least one multi-part application question per sitting.

Is the trapezoidal rule (trapezium rule) still examinable?
No. The trapezoidal rule was removed from the 9758 syllabus for examinations from 2026 onwards. [1] You do not need to practise it for the A-level papers, and questions will not ask you to approximate integrals using that method. If your school still sets trapezium rule questions, check that those worksheets target the old 9740 syllabus rather than 9758.

How should I use the graphing calculator (GC) efficiently on area and volume questions?
Use the GC to verify your final numeric answer after completing all manual working - not as a substitute for it. For area questions, enter each piece of the integral separately (e.g. ∫(f(x)-g(x),x,a,b)) and compare the GC output with your analytical result. For volumes, compute $\pi \int_a^b \bigl[f(x)\bigr]^2 , dx$ numerically to catch arithmetic slips. SEAB mark schemes award method marks for correct setup, so your written working must still show the integrand and limits explicitly even if you rely on the GC for the final value.

Other H2 Maths formula sheets

Revising more than one topic? Grab the matching one-page formula sheet:

• Functions & graphs: Functions · Graphs & Transformations · Equations & Inequalities
• Sequences & series: Sequences & Series
• Vectors: Vectors
• Calculus: Differentiation · Maclaurin Series · Integration Techniques · Definite Integrals (this page) · Differential Equations
• Statistics: Probability · Discrete Random Variables · Normal Distribution · Sampling · Hypothesis Testing · Correlation & Regression

For the official SEAB reference booklet, see the H2 Maths MF27 formula list.

Sources

• SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 5 Calculus (definite integrals, area/volume applications, improper integrals, mean value/average value): SEAB 9758 syllabus PDF

Practice Quiz

Verify that you can manage area, volume, and improper integral workflows with precise limit handling.

Quick Revision Checklist

• Compute areas between curves accurately with correct limits and integrands.
• Derive volumes of revolution using shell or disk method as appropriate.
• Handle modulus and piecewise integrals by splitting at critical points.
• Evaluate improper integrals with proper limit notation and convergence checks.

Next steps: Continue via the H2 Maths notes hub and wrap up with Topic 5.5 - Differential equations.