SPM Add Maths Integration — Tips to Score Full Marks

Integration is the second half of calculus in SPM Additional Mathematics. It is the reverse of differentiation, and — just like differentiation — it carries significant marks in both Paper 1 and Paper 2. Students who understand integration deeply can earn full marks on these questions. Here is how to approach it.

The Basic Integration Rule

The fundamental rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + c, for n ≠ -1. For constants: ∫a dx = ax + c. For a function raised to a power: ∫(ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ / [a(n+1)] + c. Always include the constant of integration +c when integrating indefinitely.

Definite Integrals

A definite integral ∫[a to b] f(x) dx gives a numerical value — no +c. Evaluate by substituting the upper limit and subtracting the result with the lower limit: [F(b)] - [F(a)]. A key point: if the curve is below the x-axis between a and b, the definite integral gives a negative value. For area calculations, take the absolute value.

Area Under a Curve

Area between a curve and the x-axis: A = |∫[a to b] y dx|. Area between a curve and the y-axis: A = |∫[c to d] x dy|. For this second type, you need to express x in terms of y. Area between two curves: A = ∫[a to b] [f(x) - g(x)] dx, where f(x) is the upper curve. Always sketch a diagram first to identify which curve is above the other.

Volume of Revolution

Volume generated when y = f(x) is rotated 360° about the x-axis: V = π∫[a to b] y² dx. Volume when rotated about the y-axis: V = π∫[c to d] x² dy. These appear in Section C of SPM Paper 2 and are worth 10 marks. Make sure you square the function correctly before integrating — this is where most errors occur.

Given dy/dx, Find y

A common SPM question type: you are given dy/dx = some expression and a point the curve passes through. Integrate to find y (don't forget +c), then substitute the given point to find the value of c. This requires both integration skill and basic substitution.

Integration Practice Strategy

As with differentiation, integration is built through practice. Spend time on each sub-type separately: basic indefinite integrals, definite integrals, area problems, volume of revolution. Past SPM papers from 2015 onwards will show you exactly how each type is examined. Focus on Paper 2 Section B and C questions for integration.