Numerical Integration Calculator

Frequently Asked Questions

When do I need numerical integration?

When an integral has no closed-form antiderivative or you only have sampled data points instead of a formula.

How do the rules differ?

The trapezoidal rule fits straight segments, the midpoint rule uses the centre height, and Simpson's rule fits parabolas - Simpson is far more accurate for smooth functions.

How does accuracy scale?

Trapezoidal and midpoint error shrinks like 1/n², Simpson's like 1/n⁴, so doubling the subintervals cuts Simpson error by about 16×.

What does the tool report?

The estimate from each rule plus the difference between them, which gives a quick sense of the remaining error.

Provided by AllCalculators.io
Free online calculators for everyday. No registration required.

Estimates for informational purposes only.

Important Disclaimer: Estimates for informational purposes only.

This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.

JavaScript is required to use the interactive calculator above. The questions and answers below remain readable without JavaScript.

How This Calculator Works

Numerical integration estimates the definite integral ∫_a^b f(x) dx - the signed area under a curve - when no clean antiderivative exists. The interval [a, b] is split into n strips of width h = (b−a)÷n. The trapezoidal rule joins endpoints with straight lines: area ≈ h[½f(a) + f(x₁) + … + ½f(b)]. The midpoint rule uses a rectangle of height f at each strip's centre. Simpson's rule fits a parabola through every pair of strips, weighting interior points 4, 2, 4, 2, … - it is exact for cubics and converges far faster. This tool ships a hand-written recursive-descent parser supporting + − * ÷ ^, x, sin, cos, tan, exp, ln, sqrt, pi and e - no JavaScript eval(), so an expression can never execute arbitrary code.

Worked Example: ∫₀^π sin x dx

The exact value is 2. With n = 8 the trapezoidal rule gives ≈ 1.9742 - it underestimates because sin is concave, so chords sit below the curve. The midpoint rule gives ≈ 2.0129, overestimating from the opposite side. Simpson's rule, blending them in a 2:1 ratio (the same combination Simpson's formula encodes), returns ≈ 2.0001 - three correct digits from just eight strips. Doubling n shrinks trapezoidal error by 4× (it is O(h²)) but Simpson's error by 16× (O(h⁴)), which is why Simpson's is the default workhorse.

Where Numerical Integration Is Used

Physics & engineering: work, centre of mass, and electrical charge from sampled data with no formula.

Statistics: areas under the normal curve (there is no elementary antiderivative of e^−x²).

Finance: expected value and option pricing integrals solved by quadrature.

Computer graphics & ML: Monte-Carlo and quadrature for lighting, rendering, and Bayesian evidence.

Common Mistakes & Edge Cases

Too few strips on an oscillatory or sharply curved function gives a poor estimate - increase n and watch the values stabilise.

Simpson's rule needs an even number of strips; this tool bumps an odd n up by one automatically.

Singularities (e.g. 1÷x near 0, or ln of a negative) produce infinities or NaN - the tool reports a domain error rather than a fake number.

Use × explicitly: write 2*x, not 2x - the parser does not assume implied multiplication.

Reversing bounds flips the sign: ∫_b^a = −∫_a^b.

Why Simpson's Rule Wins

The three rules differ in the polynomial they fit to each piece of the curve. The midpoint rule uses a flat line (degree 0) but is surprisingly accurate because its over- and under-shoots partly cancel; its error is O(h²). The trapezoidal rule fits a straight line (degree 1) yet has the same O(h²) order - with a coefficient twice as large and opposite in sign to the midpoint error. Simpson's rule exploits exactly that: the combination (2×midpoint + trapezoidal)÷3 cancels the leading error term, fitting a parabola (degree 2) and jumping to O(h⁴) accuracy. It is in fact exact for any cubic, one degree higher than its parabolic basis would suggest, thanks to a symmetry cancellation - a free extra order that makes it the default choice for smooth integrands.

Related Tools & Limits

This is a fixed-step Newton−Cotes method; adaptive quadrature (subdividing only where the curve bends sharply) and Gaussian quadrature (choosing optimal sample points rather than equally spaced ones) achieve the same accuracy with far fewer evaluations on hard integrands. For an exact symbolic answer you need a symbolic-antiderivative tool, not numerical sampling. The pairing of trapezoidal and midpoint into Simpson's is itself a glimpse of Richardson extrapolation, the natural next topic. Results are shown to 8 decimals; near singularities or for very large n, floating-point round-off eventually dominates the truncation error and accuracy stops improving.

Numerical Integration Calculator

Frequently Asked Questions

Related Math Calculators