Riemann Sum (Numerical Integration)

Approximate ∫ₐᵇ f(x) dx using N rectangles (left, right, midpoint).

Result

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How to use this calculator

Enter polynomial + bounds + N + method.

About this calculator

Riemann sums approximate definite integrals via rectangles. As N → ∞, sum → exact integral. Midpoint method has O(1/N²) error (best of three); left/right have O(1/N). Doubling N quarters midpoint error but only halves left/right error. For polynomials, exact integration is easier; Riemann is for non-elementary functions. Simpson's rule (combine pairs) is even better — O(1/N⁴). Source: Wolfram MathWorld - Riemann Sum.

Frequently asked

Best method?+

Midpoint usually wins on smooth functions. Simpson's rule even better but more complex.

Why N matters?+

Smaller rectangles = closer to true area. N=100 typically gives 4-6 decimal accuracy for polynomials.

When does it fail?+

Discontinuous functions, very oscillatory integrals. Specialized methods (adaptive quadrature) help.

Riemann vs. Simpson?+

Simpson uses parabolic interpolation, far more accurate for same N. Industry standard for numerical integration.

For real coding?+

Use library: scipy.integrate.quad, MATLAB integral(), R integrate. Don't hand-roll.

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