Simpson's Rule - Parabolic Precision

Calculus 2 · Axiom Academy

Parabolic precision: approximating integrals with parabolas instead of straight lines, for dramatically better accuracy. The Trapezoidal Rule caps each strip with a straight chord ; over a bending curve that chord leaves a gap. What if the cap could curve with the function? A parabola can — and it hugs the curve far more tightly. 2. Fitting a Parabola Through Three Points Simpson's Rule splits the interval into pairs of subintervals. Over each pair it samples three heights — the left endpoint x_0 , the midpoint x_1 , and the right endpoint x_2 — and threads a parabola through them. Three points determine exactly one parabola Because a quadratic y = ax^2 + bx + c has exactly three coefficients, three points pin it down uniquely — giving the best parabolic fit over each pair of subintervals. 3. Simpson's Rule and the 1-4-2-4-…-1 Pattern Integrating each parabola and adding the overlapping pairs produces a fixed weight pattern: endpoints get weight 1 , odd-indexed points get 4 , and the even interior points (shared between pairs) get 2 . Put all three methods on the same curve, , with n = 4 subintervals. Watch how much tighter the parabolas sit against the curve — and how the error collapses. Here is the surprise: Simpson's Rule is exactly a weighted average of the two methods you already know — the Trapezoidal and the Midpoint Rules. One part Trapezoidal, two parts Midpoint

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