Function f is known at the following points: x 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1…

Concept

The composite trapezoidal rule approximates a definite integral by joining consecutive data points with straight chords, so each strip becomes a trapezium. For n equal sub-intervals of width h, the estimate is

∫ f dx ≈ h × [ (ffirst + flast) / 2 + (sum of all interior f-values) ]

Equivalently, the two end ordinates get weight 1 and every interior ordinate gets weight 2, all multiplied by h/2.

Application

The eleven tabulated abscissae are equally spaced, so the step size is fixed:

1. Step size: h = (3.0 − 0) / 10 = 0.3, giving 10 strips across the eleven points.

2. End ordinates: f(0) = 0 and f(3.0) = 9.00, so their average is (0 + 9.00) / 2 = 4.5.

3. Interior ordinates (the nine middle values): 0.09 + 0.36 + 0.81 + 1.44 + 2.25 + 3.24 + 4.41 + 5.76 + 7.29 = 25.65.

4. Combine: integral ≈ h × (4.5 + 25.65) = 0.3 × 30.15 = 9.045.

Cross-check

The tabulated values are exactly f(x) = x2, whose exact integral over [0, 3] is x3/3 = 27/3 = 9.000. Because x2 is convex, the chords lie above the curve, so the trapezoidal estimate must be slightly larger than 9.000 — and 9.045 is indeed just above it, confirming the arithmetic. (Simpson's rule, by contrast, would return the exact 9.000 here.)

Result: ∫₀³ f(x) dx ≈ 9.045 by the trapezoidal rule.