Riemann Sum (Numerical Integration)
Approximate ∫ₐᵇ f(x) dx using N rectangles (left, right, midpoint).
Result
Riemann sum
21.332800
Exact: 21.333333 · error: 5.333e-4.
- f(x)0x³ + 1x² + 0x + 0
- [a, b][0, 4]
- N100
- Δx0.040000
- Methodmid
- Riemann sum21.33280000
- Exact21.33333333
- Error5.3333e-4
Step-by-step
- Δx = (b−a)/N = 0.0400.
- Sum f(xᵢ) × Δx for i=0..N−1.
- Result: 21.332800.
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
Midpoint usually wins on smooth functions. Simpson's rule even better but more complex.
Related calculators
Definite Integral (Polynomial)
∫ₐᵇ p(x) dx using power rule for antiderivatives.
Polynomial Derivative
d/dx(aₙxⁿ + … + a₁x + a₀) = naₙxⁿ⁻¹ + … + a₁. Power rule.
Limit Evaluator (Polynomial Ratio)
lim x→a of polynomial ratios (basic substitution + L'Hopital for 0/0).
Taylor Series Approximation
Approximate sin/cos/eˣ/ln(1+x) via Taylor polynomial of degree n.
Newton's Method Root Finder
xₙ₊₁ = xₙ − f(xₙ) / f'(xₙ). Iteratively find roots of polynomial.
Polynomial Evaluator
Evaluate polynomial p(x) = aₙxⁿ + … + a₁x + a₀ at any x. Up to degree 6.