Integration
Integration reverses differentiation and, in SPM Additional Mathematics, is applied to definite integrals, areas under curves, areas between curves and volumes of revolution. The exam expects correct antiderivatives and careful handling of limits and signs.
Core rules
∫xⁿ dx = xⁿ⁺¹/(n+1) + c for n ≠ −1. A definite integral ∫ₐᵇ f(x) dx = F(b) − F(a). The area between a curve and the x-axis is ∫|y| dx, so split the interval wherever the curve crosses the axis. The area between two curves is ∫(upper − lower) dx. The volume generated when the region between the curve and the x-axis rotates through 360° about the x-axis is π∫y² dx.
Worked example
Evaluate ∫₁² (3x² + 1) dx. The antiderivative is x³ + x. Applying the limits: (2³ + 2) − (1³ + 1) = 10 − 2 = 8.
Traps
Omitting + c in indefinite integrals; reversing the limits without changing the sign; reporting a negative value for an area instead of taking the modulus or splitting the region; squaring only part of y in the volume formula.