The Trapezoidal Rule

Calculus 2 · Axiom Academy

Approximate a definite integral by fitting trapezoids under the curve — a small change from rectangles that buys a big jump in accuracy. 1. Where Rectangles Fall Short Recall the two basic rectangle estimates. The left sum L_n uses each interval's left endpoint for the height; the right sum R_n uses the right endpoint. For an increasing function, L_n sits entirely under the curve (an underestimate) and R_n pokes above it (an overestimate). 2. The Key Insight: Tilt the Tops Instead of a flat top, connect each pair of adjacent points on the curve with a straight chord . The flat rectangle top swings up to meet the curve at both corners, turning every strip into a trapezoid that follows the curve's slope. 3. The Trapezoidal Rule Formula Split [a,b] into n strips of width , with nodes . Adding the n trapezoid areas and collecting like terms gives the rule: Equivalently, the average of the two Riemann sums: Watch the coefficients: each interior node is shared by two trapezoids, so it counts twice ; the two endpoints belong to one trapezoid each, so they count once . That is the whole reason for the pattern, and the leading comes from the averaging. Take with n = 4 , so and the nodes are . The heights there are . We can integrate this one exactly to grade each estimate against the truth, 154.2 . L_4 = 144.075 → off by 10.125 . Flat tops sit well below the rising curve. R_4 = 165.675 → off by 11.475 . Flat tops overshoot it.

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