Reading this result: Trapezoid has error O(h²), so halving the step cuts the error by ~4×. It only tracks the slope, not the curvature, so it trails Simpson badly on smooth functions. Here |error| = 2.34e-2 at n=8 (h=0.375). For smooth f, switching to Simpson typically buys several extra digits of accuracy at the same n.
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How it works
Numerical integration (quadrature) estimates a definite integral by sampling f(x) and summing simple shapes: flat-topped rectangles (Left, Right, and Midpoint Riemann sums), straight-line trapezoids, or parabolic arcs (Simpson's rule). The plot shades the exact regions each method uses. Compare the estimate against the analytic value to see the error orders in action — trapezoid converges as O(h²) while Simpson reaches O(h⁴), so Simpson buys several extra digits of accuracy for smooth functions at the same number of subintervals.
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