Simpson's Rule Calculator

This calculator helps you apply Simpson's Rule for numerical integration, ideal for students and professionals in mathematics and engineering. Enter the required parameters to compute the integral of a function over a given interval.

Lower Limit *

Upper Limit *

Number of Subintervals (even number) *

Results

Integral Value: N/A

Data Source and Methodology

All calculations are based strictly on the formulas and data provided by authoritative mathematical sources.

The Formula Explained

\[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{3} [f(x_0) + 4 \sum_{i=1,3,5}^{n-1} f(x_i) + 2 \sum_{i=2,4,6}^{n-2} f(x_i) + f(x_n)] \]

Glossary of Variables

Lower Limit (a): The start of the interval.
Upper Limit (b): The end of the interval.
Subintervals (n): The number of divisions in the interval, which must be even.
Function (f(x)): The function to integrate.

Frequently Asked Questions (FAQ)

What is Simpson's Rule?

Simpson's Rule is a method for numerical integration, the numerical approximation of definite integrals.

Why must the number of subintervals be even?

Simpson's Rule requires an even number of subintervals because it uses pairs of subintervals to approximate the curve with parabolas.

What types of functions work best with Simpson's Rule?

Simpson's Rule works best with smooth, continuous functions without significant oscillations.

How accurate is Simpson's Rule?

The accuracy depends on the function and the number of subintervals; increasing subintervals generally improves accuracy.

Can I use Simpson's Rule for improper integrals?

Simpson’s Rule is not suitable for improper integrals without modifications.

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Audit: Complete

Formula (LaTeX) + variables + units

This section shows the formulas used by the calculator engine, plus variable definitions and units.

Formula (extracted LaTeX)

\[','\]

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Formula (extracted LaTeX)

\[\int_{a}^{b} f(x) \, dx \approx \frac{h}{3} [f(x_0) + 4 \sum_{i=1,3,5}^{n-1} f(x_i) + 2 \sum_{i=2,4,6}^{n-2} f(x_i) + f(x_n)]\]

\int_{a}^{b} f(x) \, dx \approx \frac{h}{3} [f(x_0) + 4 \sum_{i=1,3,5}^{n-1} f(x_i) + 2 \sum_{i=2,4,6}^{n-2} f(x_i) + f(x_n)]

Formula (extracted text)

\[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{3} [f(x_0) + 4 \sum_{i=1,3,5}^{n-1} f(x_i) + 2 \sum_{i=2,4,6}^{n-2} f(x_i) + f(x_n)] \]

Variables and units

No variables provided in audit spec.

Sources (authoritative):

NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures
FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/

Changelog

Version: 0.1.0-draft
Last code update: 2026-01-19

0.1.0-draft · 2026-01-19

Initial audit spec draft generated from HTML extraction (review required).
Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
Confirm sources are authoritative and relevant to the calculator methodology.

Verified by Ugo Candido on 2026-01-19
Profile · LinkedIn

, ', svg: { fontCache: 'global' } };

Simpson's Rule Calculator

Lower Limit *

Upper Limit *

Number of Subintervals (even number) *

Results

Integral Value: N/A

Data Source and Methodology

All calculations are based strictly on the formulas and data provided by authoritative mathematical sources.

The Formula Explained

\[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{3} [f(x_0) + 4 \sum_{i=1,3,5}^{n-1} f(x_i) + 2 \sum_{i=2,4,6}^{n-2} f(x_i) + f(x_n)] \]

Glossary of Variables

Lower Limit (a): The start of the interval.
Upper Limit (b): The end of the interval.
Subintervals (n): The number of divisions in the interval, which must be even.
Function (f(x)): The function to integrate.

Frequently Asked Questions (FAQ)

What is Simpson's Rule?

Simpson's Rule is a method for numerical integration, the numerical approximation of definite integrals.

Why must the number of subintervals be even?

Simpson's Rule requires an even number of subintervals because it uses pairs of subintervals to approximate the curve with parabolas.

What types of functions work best with Simpson's Rule?

Simpson's Rule works best with smooth, continuous functions without significant oscillations.

How accurate is Simpson's Rule?

The accuracy depends on the function and the number of subintervals; increasing subintervals generally improves accuracy.

Can I use Simpson's Rule for improper integrals?

Simpson’s Rule is not suitable for improper integrals without modifications.

```

Audit: Complete

Formula (LaTeX) + variables + units

This section shows the formulas used by the calculator engine, plus variable definitions and units.

Formula (extracted LaTeX)

\[','\]

','

Formula (extracted LaTeX)

\[\int_{a}^{b} f(x) \, dx \approx \frac{h}{3} [f(x_0) + 4 \sum_{i=1,3,5}^{n-1} f(x_i) + 2 \sum_{i=2,4,6}^{n-2} f(x_i) + f(x_n)]\]

\int_{a}^{b} f(x) \, dx \approx \frac{h}{3} [f(x_0) + 4 \sum_{i=1,3,5}^{n-1} f(x_i) + 2 \sum_{i=2,4,6}^{n-2} f(x_i) + f(x_n)]

Formula (extracted text)

\[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{3} [f(x_0) + 4 \sum_{i=1,3,5}^{n-1} f(x_i) + 2 \sum_{i=2,4,6}^{n-2} f(x_i) + f(x_n)] \]

Variables and units

No variables provided in audit spec.

Sources (authoritative):

NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures
FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/

Changelog

Version: 0.1.0-draft
Last code update: 2026-01-19

0.1.0-draft · 2026-01-19

Initial audit spec draft generated from HTML extraction (review required).
Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
Confirm sources are authoritative and relevant to the calculator methodology.

Verified by Ugo Candido on 2026-01-19
Profile · LinkedIn

]], displayMath: [['\\[','\\]']] }, svg: { fontCache: 'global' } };, svg: { fontCache: 'global' } };

Simpson's Rule Calculator

Lower Limit *

Upper Limit *

Number of Subintervals (even number) *

Results

Integral Value: N/A

Data Source and Methodology

All calculations are based strictly on the formulas and data provided by authoritative mathematical sources.

The Formula Explained

\[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{3} [f(x_0) + 4 \sum_{i=1,3,5}^{n-1} f(x_i) + 2 \sum_{i=2,4,6}^{n-2} f(x_i) + f(x_n)] \]

Glossary of Variables

Lower Limit (a): The start of the interval.
Upper Limit (b): The end of the interval.
Subintervals (n): The number of divisions in the interval, which must be even.
Function (f(x)): The function to integrate.

Frequently Asked Questions (FAQ)

What is Simpson's Rule?

Simpson's Rule is a method for numerical integration, the numerical approximation of definite integrals.

Why must the number of subintervals be even?

Simpson's Rule requires an even number of subintervals because it uses pairs of subintervals to approximate the curve with parabolas.

What types of functions work best with Simpson's Rule?

Simpson's Rule works best with smooth, continuous functions without significant oscillations.

How accurate is Simpson's Rule?

The accuracy depends on the function and the number of subintervals; increasing subintervals generally improves accuracy.

Can I use Simpson's Rule for improper integrals?

Simpson’s Rule is not suitable for improper integrals without modifications.

```

Audit: Complete

Formula (LaTeX) + variables + units

This section shows the formulas used by the calculator engine, plus variable definitions and units.

Formula (extracted LaTeX)

\[','\]

','

Formula (extracted LaTeX)

\[\int_{a}^{b} f(x) \, dx \approx \frac{h}{3} [f(x_0) + 4 \sum_{i=1,3,5}^{n-1} f(x_i) + 2 \sum_{i=2,4,6}^{n-2} f(x_i) + f(x_n)]\]

\int_{a}^{b} f(x) \, dx \approx \frac{h}{3} [f(x_0) + 4 \sum_{i=1,3,5}^{n-1} f(x_i) + 2 \sum_{i=2,4,6}^{n-2} f(x_i) + f(x_n)]

Formula (extracted text)

\[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{3} [f(x_0) + 4 \sum_{i=1,3,5}^{n-1} f(x_i) + 2 \sum_{i=2,4,6}^{n-2} f(x_i) + f(x_n)] \]

Variables and units

No variables provided in audit spec.

Sources (authoritative):

NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures
FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/

Changelog

Version: 0.1.0-draft
Last code update: 2026-01-19

0.1.0-draft · 2026-01-19

Initial audit spec draft generated from HTML extraction (review required).
Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
Confirm sources are authoritative and relevant to the calculator methodology.

Verified by Ugo Candido on 2026-01-19
Profile · LinkedIn