Numerical Integral Calculator – Estimate ∫

The estimator attacks definite integrals two ways. Symbolically, antiderivatives close the problem exactly: ∫₀¹ x² dx = [x³/3]₀¹ = 1/3, and ∫₀^π sin x dx = [−cos]₀^π = 2. Numerically, when no antiderivative exists (or one is unknown), the trapezoid rule and Simpson’s rule approximate from sampled values with errors printed for each: on ∫₀¹ x⁴ dx with n = 4 the trapezoid rule returns 0.2207 versus the true 0.2 (10.3% high), while Simpson at the same nodes gives 0.20052 — 0.26% high, with its h⁴ error shrinking 16× per refinement. The calculator shows node tables, weights, both estimates, and the exact value when its library recognizes the integrand — so students see convergence, not just answers. Absolute and relative errors quantify which rule to trust at each n.

1. 1

   Enter f(x) and bounds

   The integrand over [a, b]

2. 2

   Try the antiderivative

   Known forms return exact values like 1/3

3. 3

   Run the numeric rules

   Trapezoid and Simpson estimates with node tables

4. 4

   Compare errors

   Absolute and relative errors rank the methods

• 1

  Antiderivative F gives exact F(b) − F(a)

• 2

  Simpson beats trapezoid by two error orders on smooth f

• 3

  Error shrinks 4× (trapezoid) or 16× (Simpson) per doubling of n

• Forgetting to evaluate F at both bounds and subtract

• Dropping the constant of integration in definite settings (it cancels anyway)

• Using Simpson with odd n

• Believing numeric rules handle discontinuities gracefully