Numerical Integration Approximator

Math Calculus • Integrals

Written by STEM Calculators Team Published December 28, 2025 Updated February 24, 2026

8. Numerical Integration Approximator

Approximates definite integrals with Trapezoidal, Midpoint, Simpson, Adaptive Simpson, and Monte Carlo. Shows partition visual, error estimates, and compares vs exact (when recognized) or a high-accuracy reference.

Inputs

Function

Use x. Allowed: + − * / ^ (or **), parentheses, pi, e, sin/cos/tan/sec/csc/cot, exp, ln/log, sqrt, abs, min/max.

Lower bound \(a\)

Upper bound \(b\)

Method

Trapezoid/Midpoint/Simpson use \(n\) subintervals; Adaptive Simpson uses a tolerance; Monte Carlo uses \(N\) samples.

n intervals

For Simpson, \(n\) is forced to even (if odd, it becomes n+1).

Options

Show steps Show partition lines Show method approximation Shade true area \([a,b]\) Grid

Presets

Click a preset to load and evaluate.

Ready

Graph

Drag to pan • wheel/pinch to zoom. Partition and approximation are visualized inside \([a,b]\).

Center \(c\)

Half-width \(w\)

Legend

\(f(x)\) true shaded \([a,b]\) method approx partition

x: 0, y: 0, zoom(px/unit): 120

Result

Enter \(f(x)\), bounds \(a,b\), choose a method, then click Evaluate.

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8. Numerical Integration Approximator — Theory

Numerical integration approximates the definite integral

\[ I=\int_a^b f(x)\,dx \]

by sampling \(f\) on a partition of \([a,b]\). This calculator visualizes the partition and the geometric approximation.

1) Composite Trapezoidal Rule

\[ h=\frac{b-a}{n},\qquad T_n=h\left(\frac{f(a)+f(b)}{2}+\sum_{i=1}^{n-1} f(a+ih)\right) \]

For smooth \(f\), the global error is typically \(O(h^2)\).

\[ \left|I-T_n\right|\approx \frac{|T_{2n}-T_n|}{3}. \]

2) Composite Midpoint Rule

\[ h=\frac{b-a}{n},\qquad M_n=h\sum_{i=0}^{n-1} f\!\left(a+\left(i+\tfrac12\right)h\right) \]

For smooth \(f\), the global error is typically \(O(h^2)\).

\[ \left|I-M_n\right|\approx \frac{|M_{2n}-M_n|}{3}. \]

3) Composite Simpson’s Rule

Simpson’s rule fits a parabola through points in pairs of subintervals, so it requires even \(n\).

\[ h=\frac{b-a}{n},\; n\text{ even},\qquad S_n=\frac{h}{3}\left(f(a)+f(b)+4\!\!\sum_{\text{odd }i} f(a+ih)+2\!\!\sum_{\text{even }i} f(a+ih)\right). \]

For smooth \(f\), Simpson’s global error is typically \(O(h^4)\).

\[ \left|I-S_n\right|\approx \frac{|S_{2n}-S_n|}{15}. \]

4) Adaptive Simpson

Adaptive Simpson recursively subdivides intervals until an estimated error falls below a user tolerance \(\varepsilon\).

\[ S[a,b]=\frac{b-a}{6}\left(f(a)+4f\!\left(\tfrac{a+b}{2}\right)+f(b)\right). \]

5) Monte Carlo

Monte Carlo draws random samples \(X_k\sim\mathrm{Uniform}(a,b)\).

\[ I \approx (b-a)\cdot \frac{1}{N}\sum_{k=1}^N f(X_k),\qquad \text{Std. error}\approx (b-a)\cdot\frac{\sigma_f}{\sqrt{N}}. \]

6) Exact vs reference

If no exact form is recognized, the tool computes a high-accuracy reference value to estimate the true error of your chosen method.

Frequently Asked Questions

Which numerical integration method should I use for the best accuracy?

For smooth functions, composite Simpson often gives high accuracy with relatively small n, while Adaptive Simpson refines automatically to meet a tolerance. Monte Carlo is useful when you want a sampling-based estimate, but it typically converges more slowly than Simpson-type rules.

Why does Simpson's rule require an even number of subintervals?

Composite Simpson's rule applies a quadratic fit over pairs of subintervals, so n must be even to group the partition into pairs. If n is odd, it must be increased by 1 to make the method valid.

What does the tolerance mean in Adaptive Simpson integration?

Tolerance is the target error threshold used to decide when to stop subdividing intervals. Smaller tolerance usually increases accuracy but can require more recursion and computation.

What does the Monte Carlo seed do?

The seed controls the random number stream used to draw samples on [a,b]. Using the same seed makes the Monte Carlo estimate reproducible for the same inputs.

What does it mean to compare against an exact value or a high-accuracy reference?

If the tool recognizes a closed-form exact value, it can compute the true error directly. If no exact form is recognized, it computes a high-accuracy reference value and uses it to estimate the true error of the chosen method.

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Frequently Asked Questions

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