Approximating ∫₀² e⁻ˣ²dx

Calculus 2 · Axiom Academy

Approximate this integral four different ways and discover which method is most accurate. The integral has no elementary antiderivative, so we approximate it numerically. We compare the Left and Right Riemann sums, the Trapezoidal Rule, and Simpson's Rule, all with n = 4 subintervals. Nice work — you compared all the major numerical-integration methods on the same integral. Here's what stands out: Simpson's Rule is superior: with only 4 subdivisions it reached 0.02% error, versus about 28% for the Riemann sums. The Trapezoidal Rule is the average: T_4 = (L_4 + R_4)/2 , giving moderate accuracy at 0.16% error. Why Simpson's wins: it fits parabolic arcs instead of straight lines, matching a curved function like e^ -x^2 much better. Trade-offs matter: Simpson's needs an even n and a bit more arithmetic, but the accuracy gain is usually worth it. Left vs. Right: for a decreasing function the Left sum overestimates and the Right sum underestimates, so the truth lies between them (the reverse holds for increasing functions). For integrals with no closed-form answer, Simpson's Rule at modest n often gives excellent accuracy — which is why it is a go-to method across science and engineering.

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