Simpson’s 1/3 Rule & Trapezoidal Rule Definite Integral Approximation Calculator

Compare Composite Simpson’s 1/3 Rule and Trapezoidal Rule against exact definite integration.

Simpson’s 1/3 Rule Integral Approximation
6.389 integral
Composite Trapezoidal Approximation 6.422 integral
Exact Analytical Definite Integral 6.389 integral
Trapezoidal Minus Exact Error 0.033 |err|
Inputs
⚡ Scenarios:
a
Lower integration bound a.
b
Upper integration bound b.
k
Exponent parameter k in f(x) = e^(k·x).
subintervals
Number of even subintervals N (must be ≥2 even integer).
Results & Graph
Simpson’s 1/3 Rule Integral Approximation
6.389 integral
Composite Trapezoidal Approximation
6.422 integral
Exact Analytical Definite Integral
6.389 integral
Trapezoidal Minus Exact Error
0.033 |err|

Simpson’s 1/3 Rule Integral Approximation

Graph 1 • Simpson’s 1/3 & Trapezoidal Rule • Domain Architecture & Live Telemetry Maps Lower Limit A, Upper Limit B, Exponent Growth K through Simpson’s 1/3 & Trapezoidal Rule Formula.
Simpson’s 1/3 Rule Integral Ap: 6.389 integral Simpson’s 1/3 & Trapezoidal Rule • Invariants & Metrics Simpson’s 1/3 Rule Integral 6.389 integral Composite Trapezoidal Approx 6.422 integral Exact Analytical Definite In 6.389 integral Trapezoidal Minus Exact Erro 0.033 |err|
Graph 2 • Simpson’s 1/3 Rule Integral Approximation Across Lower Limit A Sweep (40% → 160%) Evaluates Simpson = (h / 3) × [ f(x_0) + 4·Σf(x_odd) + 2·Σf(x_even) + f(x_n) ]; across 7 operating points (current input highlighted).
0 integr 49.1k integr 98.2k integr 0.549 4a 4.05 6a 29.96 8a 221.4 10a 1.6k 12a 12.1k 14a 89.3k 16a
Graph 3 • Joint Sensitivity: Lower Limit A × Upper Limit B Compares Simpson’s 1/3 Rule Integral Approximation when Upper Limit B is shifted +25% (solid) vs. −25% (dashed).
0 integr 2.4k integr 4.9k integr --- Upper Limit B @ 75% 11.02 -30% 49.39 -15% 221.4 +0% 992.1 +15% 4.4k +30%
Summary: Across 8 subintervals (h=0.25), Simpson’s 1/3 Rule yields 6.389194 (exact: 6.389056) vs. Trapezoidal Rule 6.4223.
Summary: Across 8 subintervals (h=0.25), Simpson’s 1/3 Rule yields 6.389194 (exact: 6.389056) vs. Trapezoidal Rule 6.4223.
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Formula & Mathematical Methodology: Simpson’s 1/3 & Trapezoidal Rule Formula

Simpson = (h / 3) × [ f(x0) + 4·Σf(xodd) + 2·Σf(xeven) + f(xn) ]
Trapezoid = (h / 2) × [ f(x0) + 2·Σf(xi) + f(xn) ]
Simpson = (h / 3) × [ f(x_0) + 4·Σf(x_odd) + 2·Σf(x_even) + f(x_n) ]; Trapezoid = (h / 2) × [ f(x_0) + 2·Σf(x_i) + f(x_n) ]
  • lowerLimitA: Lower Limit A (a)
  • upperLimitB: Upper Limit B (b)
  • exponentGrowthK: Exponent Growth K (k)
  • evenSubintervalsN: Even Subintervals N (subintervals)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Simpson’s 1/3 & Trapezoidal Rule

Compare Composite Simpson’s 1/3 Rule and Trapezoidal Rule against exact definite integration.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: Simpson = (h / 3) × [ f(x_0) + 4·Σf(x_odd) + 2·Σf(x_even) + f(x_n) ]; Trapezoid = (h / 2) × [ f(x_0) + 2·Σf(x_i) + f(x_n) ].
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Simpson’s 1/3 & Trapezoidal Rule via Simpson = (h / 3) × [ f(x_0) + 4·Σf(x_odd) + 2·Σf(x_even) + f(x_n) ]; Trapezoid = (h / 2) × [ f(x_0) + 2·Σf(x_i) + f(x_n) ].

Frequently Asked Questions

How does the Simpson’s 1/3 Rule & Trapezoidal Rule Definite Integral Approximation Calculator compute the solution?

It evaluates Simpson = (h / 3) × [ f(x_0) + 4·Σf(x_odd) + 2·Σf(x_even) + f(x_n) ]; Trapezoid = (h / 2) × [ f(x_0) + 2·Σf(x_i) + f(x_n) ] directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Numerical Analysis & Calculus

Simpson’s 1/3 Rule & Trapezoidal Rule Definite Integral Approximation Calculator: inputs, method, and how to read the result

Compare Composite Simpson’s 1/3 Rule and Trapezoidal Rule against exact definite integration.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Lower Limit A: Lower integration bound a.
  • Upper Limit B: Upper integration bound b.
  • Exponent Growth K: Exponent parameter k in f(x) = e^(k·x).
  • Even Subintervals N: Number of even subintervals N (must be ≥2 even integer).

What the calculation includes

With the default values shown, Simpson’s 1/3 Rule Integral Approximation is 6.389 integral. The calculation follows Simpson’s 1/3 & Trapezoidal Rule Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Lower Limit A from 0 a to 0.01 a changes Simpson’s 1/3 Rule Integral Approximation from 6.389 integral to 6.379 integral. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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