Which Riemann sum rule should I use?

Left and right sums are quickest for monotone functions but can overshoot when f increases or decreases. The trapezoid rule is O(h^2) accurate and works well for smooth curves, while Simpson's rule reaches O(h^4) accuracy if f is twice differentiable. Use midpoint when you want a single-rectangle interpretation with better symmetry.

Riemann Sums Explorer — Left, Right, Trapezoid, Simpson

Q: Why must Simpson's rule use an even n?

Simpson's rule fits parabolas through adjacent point triplets, so the interval must be split into an even number of subintervals. The calculator automatically bumps n to the next even number and notes the change in the step log.

Type analytic expressions such as sin(x), exp(-x^2), x^3 - 2x, or combinations that call ln, abs, sgn, and constants like pi.

Need a quick class demo? Toggle the fill option to emphasise signed area, compare how each rule converges as n grows, and use the CSV or LaTeX buttons to build worksheets instantly.

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f(x) Approximation Simpson segments

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Approx Sₙ

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Reference integral

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Absolute error

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Relative error

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How it's calculated

Teacher notes

FAQ

Which Riemann sum rule should I pick?

Left/right sums follow the orientation of rectangles, which is great for quick estimates but can over- or undershoot when f is monotone. The trapezoid rule is second-order accurate and balances speed with precision. Simpson's rule reaches fourth-order accuracy on smooth functions, while midpoint splits the difference with a symmetric single-rectangle view.

Why must Simpson's rule use an even n?

Simpson's rule joins point triplets with quadratics, so the interval must break into an even number of subintervals. The explorer automatically increases n by 1 when needed and highlights the adjustment in the steps.

Inputs & options

Visualisation

Result

How it's calculated

FAQ

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