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The area between two curves integrates the vertical gap: ∫(upper − lower) dx. Using the absolute difference handles intervals where the curves swap places.
Area between curves
A = ∫ₐᵇ |f(x) − g(x)| dx
∫(x − x²) dx = 1/2 − 1/3 = 1/6 ≈ 0.1666667.
Split the interval at the crossing points, or integrate the absolute difference as done here.