H2 Maths Definite Integrals Formula Sheet | Area & Volume
H2 Maths Definite Integrals Formula Sheet | Area & Volume
Study guide//Updated 21 Aug 2026
Reviewed by
Marcus Pang·Managing Director (Maths)
H2 Maths definite integrals formula sheet: definite-integral properties, area under a curve, area between curves, and volume of revolution about the x- and y-axes - every key re...
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.
Properties: linearity, additivity over intervals, and reversal of limits introducing a negative sign.
The definite integral of f from a to b is the signed area between the curve y = f(x) and the x-axis over that interval. Area above the axis counts as positive and area below as negative, so a curve that dips below the axis needs the interval split at each x-intercept.
Determine intersection points by solving ytop=ybottom.
Area setup checkpoint
Before writing the integral, decide what the answer represents.
Question wording
Setup to write
What to check on the sketch
"Find the value of the integral"
Keep the signed integral ∫abf(x)dx.
Parts below the x-axis stay negative.
"Find the area under the curve"
Split where the curve crosses the x-axis and add positive pieces.
Mark every x-intercept inside the interval.
"Find the area between two curves"
Use top minus bottom, splitting at crossings if the order changes.
Label which curve is above on each subinterval.
"Find the area using horizontal strips"
Use right minus left with respect to y.
Rewrite the boundaries as x=⋯.
Misconception check: area is not always the same as ∫abf(x)dx. The definite integral is signed area; a physical region uses positive pieces.
Example -- Area enclosed
Find area between y=x2 and y=2x+3.
Intersections solve x2=2x+3 ⇒ x=−1,3.
Area =∫−13(2x+3−x2)dx=[x2+3x−3x3]−13=332.
Volumes of Revolution
About x-axis: V=π∫aby2dx.
About y-axis: V=π∫cdx2dy.
Shell method (if convenient): V=2π∫abxydx.
Volume axis checkpoint
Before calculating volume, decide whether your slice is perpendicular or parallel to the axis of rotation.
Rotation clue
First slice choice
Integral setup
Trap to avoid
Region under y=f(x) about the x-axis
Vertical slice, disk radius y.
V=π∫ab[f(x)]2dx.
Do not integrate f(x) without squaring the radius.
Region between y=f(x) and y=g(x) about the x-axis
Vertical slice, washer radii from the axis.
V=π∫ab(R2−r2)dx
Region about the y-axis using disks
Horizontal slice, radius x.
Rewrite as x=h(y), then use V=π∫cd[h(y)]2dy
Region about the y-axis using shells
Vertical slice parallel to the axis.
V=2π∫ab(radius)(height)dx.
The shell radius is distance from the y-axis, usually x
Worked check: Rotating y=x, 0≤x≤4, about the x-axis uses radius y. Since y2=x, the setup is V=π∫04xdx, not π∫04xdx.
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=x from x=0 to x=4 revolved about x-axis:
V=π∫04xdx=π[2x2]04=8π.
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=2π∫02x(2−x),dx=2π[x2−3x3]02=2π(4−38)=38π.
Mean Value and Average
Average value of f(x) on [a,b]: fˉ=b−a1∫abf(x)dx.
For probability density functions, integrals compute probabilities and expected values.
Mean-value interpretation checkpoint
Before substituting into the formula, decide what the integral is averaging.
Question clue
Integral meaning
What to divide by
Common trap
Average value of a function on [a,b]
Total signed accumulation over the interval
Interval length b−a
Giving only the integral without dividing by the interval length.
Average height of a curve above the x-axis
Area under the curve spread evenly across the base
Horizontal width
Treating the average height as the maximum height.
Probability density over an interval
Probability comes from area under the density curve
Do not divide unless the question asks for conditional average or expectation
Dividing a probability by interval length just because an integral appeared.
Expected value of a continuous random variable
Weighted average of values using density
Use ∫xf(x)dx, not ∫f(x)dx alone
Forgetting the extra factor x
Worked check: if f(x)=x2 on [0,3], the integral gives total accumulation ∫03x2dx=9. The average value is 9/(3−0)=3, so the rectangle of height 3 over width 3 has the same area as the curve.
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)=sinx on [0,π] is
fˉ=π−01∫0πsinx,dx=π1[−cosx]0π=π2.
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.
Question clue
Breakpoint to find first
Integral setup
Common trap
∣x−a∣ on an interval crossing a
x=a.
Use a−x to the left and x−a to the right.
Keeping x−a on both sides and getting a negative area.
∣f(x)∣
Roots of f(x)=0 inside the bounds.
Split at every root, then choose the sign from a test point.
Only splitting at the endpoints.
Piecewise-defined f(x)
The given joining points and any sign-change points.
Integrate each formula over its own interval.
Applying one formula outside its stated interval.
Area involving a modulus graph
Sketch the V-shape or reflected part first.
Add positive area pieces after splitting.
Treating the signed integral as the physical area without checking the graph.
Worked check: for ∫−23∣x−1∣dx, the sign changes at x=1. On [−2,1], x−1≤0, so ∣x−1∣=1−x. On [1,3], x−1≥0, so ∣x−1∣=x−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.
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 ∫a∞f(x),dx, evaluate limt→∞∫atf(x)dx.
For ∫−∞bf(x),dx, use limt→−∞∫tbf(x)dx
Example -- Tail integral
∫1∞x21dx=t→∞lim[−x1]1t=t→∞lim(−t1+1)=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.
Each limit diverges, so the original integral diverges.
Convergence quick checks
Benchmark integrals: ∫1∞xp1,dx converges for p>1 and diverges otherwise. ∫01xp1,dx converges for p<1 and diverges otherwise.
Use comparison: if 0≤f(x)≤g(x) for large x and ∫g(x),dx
Always state the limit value or declare divergence explicitly; never leave an improper integral without a convergence justification.
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 π 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.
Sign errors with area below the x-axis: ∫abf(x),dx gives a negative value when the curve lies below the x-axis, but area is always non-negative. Examiners expect you to take the modulus of each signed piece separately before summing - writing Area=∫abf(x),dx
Confusing area between curves with the net integral: The net integral ∫ab(f(x)−g(x)),dx can cancel when the curves cross within [a,b]
Wrong formula for volumes of revolution: Revolving about the x-axis uses V=π∫ab[f(x)]2,dx; revolving about the y-axis requires V=π∫cd[x(y)]2,dy
Skipping the sketch before computing area or volume: A rough sketch takes under a minute and immediately reveals which function is on top, where the region is bounded, and whether horizontal or vertical slices are more convenient. Candidates who skip this step routinely set up the wrong integrand or limits and cannot recover the marks for the correct method.
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 π∫ab[f(x)]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:
Subtract squared radii, not the original functions first unless the square is kept.
.
Do not keep dx limits after switching to horizontal slices.
, not the curve height.
.
. If both tails are infinite, split at a convenient point and handle each limit separately.
converges, then
∫f(x),dx
converges.
when part of the region dips below the axis will lose all accuracy marks for that part.
. Always sketch first, locate all intersection points, and split the integral at each crossing so you integrate
∣f(x)−g(x)∣
across the full region.
(or the shell formula). Mixing these up - for example, applying the x-axis formula to a y-axis revolution - is a conceptual error that voids the entire volume calculation.
