MM7: Integral Calculus, Areas and Function Symmetry
Official Cambridge & Imperial ESAT specification requirements for Integral Calculus, Areas and Function Symmetry on the Mathematics 2 paper. Understand the syllabus boundaries, spot distractor traps engineered by examiners, and master the 2-step Sub-90 shortcuts that compress solve time well below the 88.8-second cutoff.
Official Syllabus Subtopics
Requirements specified in the UAT-UK framework
- 1Indefinite and definite integration of polynomial/rational powers: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)
- 2Fundamental Theorem of Calculus: ∫ f(x) dx from a to b = F(b) − F(a)
- 3Area enclosed between two intersecting curves: ∫ (y(upper) − y(lower)) dx
- 4Odd function integral symmetry over [-a, a]: ∫ f(odd)(x) dx over [−a, a] = 0
- 5Even function integral symmetry over [-a, a]: ∫ f(even)(x) dx over [−a, a] = 2 ∫ f(x) dx over [0, a]
- 6Linear integral transformations and boundary substitution
Sub-90 Exam Speed Tips
How top candidates solve this spec under the 88.8s clock
If f(x) is an odd function (f(-x) = -f(x)), its integral over symmetric bounds [-a, a] is 0 without doing any algebra.
Always sketch the curves to find all intersection points and determine which curve lies above the other.
Examiners engineer distractor options using common algebraic expansions and sign slips. Always test structural ratios or invariants before calculating arithmetic values.
Tested Topics in Specification MM7
Master the individual question patterns and Sub-90 shortcuts for this specification.
Definite Integration of Rational Powers (xⁿ, n ≠ -1) & Area Enclosed Between Curves
For the area enclosed between a parabola y = ax² + bx + c and a line intersecting at x = alpha and x = beta, use Archimedes' 1/6 shortcut: Area = |a|/6 * (beta - alpha)³ in 10 seconds!
Odd/Even Function Integral Symmetry over [-a, a] & Linear Integral Transformations
Over symmetric limits [-a, a], cross out all odd powers (x, x³, x⁵) immediately (= 0) and double the integral of even powers from [0, a]. For vertical shift +c, add rectangle area c*(b-a).
Ready to Master Specification MM7 Under Exam Conditions?
Practice timed questions under the authentic Pearson VUE interface with live split stopwatch pacing, dry-wipe scratchpad, and distractor forensics.