Integral Calculator

Calculate definite integrals of any function using Simpson's rule, trapezoid rule, and Riemann sums. Includes indefinite antiderivative reference table.

Definite Integral ∫ₐᵇ f(x)dx (numerical approx.)
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Absolute Area (unsigned) —
Average Value of f on [a,b] —
Extended More scenarios, charts & detailed breakdown ▾
Simpson's 1/3 Rule (numerical approx.)
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Professional Full parameters & maximum detail ▾

Integration Methods

Trapezoid Rule (numerical approx.) —
Simpson's 1/3 Rule (numerical approx.) —
Simpson's 3/8 Rule (numerical approx.) —
5-point Gauss-Legendre (numerical approx.) —

Error Estimate

Error estimate |Simp − Gauss| —

How to Use This Calculator

  1. Enter f(x) using standard notation (e.g. x^2, sin(x)).
  2. Set the lower bound a and upper bound b.
  3. The integral is computed using Simpson's 1/3 rule with 1000 subintervals.
  4. Use the Riemann Sum tab to compare left, right, and midpoint approximations.
  5. The Professional tab compares four numerical methods with error estimates.

Formula

Simpson's 1/3: ∫ₐᵇ ≈ (h/3)[f(x₀)+4f(x₁)+2f(x₂)+…+f(xₙ)], h=(b−a)/n

Average value: f̄ = (1/(b−a)) ∫ₐᵇ f(x)dx

Example

∫₀¹ x² dx: exact = 1/3 ≈ 0.33333. Simpson n=1000 gives 0.33333333.

Frequently Asked Questions

  • The definite integral ∫ₐᵇ f(x)dx gives the net signed area under the curve y=f(x) from x=a to x=b. Positive area is above the x-axis, negative is below.
  • Simpson's 1/3 rule approximates the integral by fitting parabolas through pairs of subintervals: ∫≈(h/3)[f(x₀)+4f(x₁)+2f(x₂)+4f(x₃)+…+f(xₙ)]. It is much more accurate than the trapezoid rule for smooth functions.
  • A Riemann sum is a finite approximation using rectangles. As the number of rectangles n→∞, the Riemann sum converges to the exact definite integral.
  • For integrals with infinite bounds, use the Improper Integral Calculator which substitutes large finite values (e.g. 10⁶) and checks convergence.
  • The average value of f on [a,b] is (1/(b−a)) × ∫ₐᵇ f(x)dx. The Professional tab computes this automatically.

Related Calculators

📐 Derivative Calculator ∫ₐᵇ Definite Integral Calculator ∞ Improper Integral Calculator
Sources & References (5) ▾
  1. Integration — Paul's Online Math Notes — Lamar University
  2. Integral Calculus — Khan Academy
  3. MIT OCW 18.01 Integration — MIT
  4. Integral — Wolfram MathWorld — Wolfram MathWorld
  5. NIST Digital Library of Mathematical Functions — NIST