Integration
Use antiderivatives, definite integrals, the Fundamental Theorem, range combinations, trapezia and simple differential equations.
How to prepare for the current TMUA
Secure the shared mathematical specification, practise application in unfamiliar settings, then make every Paper 2 implication and proof step explicit.
Core concepts
Concept 1
Integrate rational powers and determine constants from conditions.
Exam cue: Simplify the integrand before applying the power rule.
Concept 2
Distinguish signed definite integrals from geometric areas.
Exam cue: Mark where a graph lies below the axis before interpreting area.
Concept 3
Combine adjacent integrals and use trapezia or dy/dx = f(x) models.
Exam cue: Use limits and interval direction carefully when combining integrals.
Risk pitfalls and guardrails
Forgetting the constant of integration.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Calling a negative definite integral a negative geometric area.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Applying the power rule at the excluded exponent.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Memory anchors
Power integral
∫xⁿdx = xⁿ⁺¹/(n+1) + C for n ≠ −1.
Upper minus lower
Evaluate an antiderivative at the upper limit then subtract the lower.
Signed area
Below-axis contributions are negative in a definite integral.
Fundamental link
Differentiation and integration undo one another under the theorem's conditions.
Contiguous ranges
Adjacent intervals can be joined when direction is consistent.
Trapezium judgement
Curve shape determines whether the straight-edged estimate is high or low.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
A curve has ordinate y=3x²+1. What signed area lies between it and the x-axis from x=0 to x=2?
A function has derivative 4x−3 and f(2)=7. What is f(0)?
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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