Calculator
Formula Used
Composite Simpson error uses the fourth derivative maximum. The bound is:
|E| ≤ ((b - a) / 180) × h4 × max|f''''(x)|
Since h = (b - a) / n, the same rule is:
|E| ≤ max|f''''(x)| × (b - a)5 / (180n4)
Actual error is |Exact integral - Simpson approximation|. Percent error is actual error divided by |Exact integral|, then multiplied by 100.
How to Use This Calculator
- Enter the lower and upper integration limits.
- Enter an even number of subintervals.
- Enter the largest expected value of |f''''(x)|.
- Add a safety factor if your derivative estimate is uncertain.
- Enter exact and Simpson values when you want actual error.
- Enter tolerance to check whether the approximation is acceptable.
- Press submit to view the result above the form.
- Use CSV or PDF buttons to download the report.
Example Data Table
| Case | a | b | n | Max |f''''(x)| | Tolerance | Purpose |
|---|---|---|---|---|---|---|
| Polynomial check | 0 | 4 | 10 | 12 | 0.01 | Bound estimate |
| Small interval | 1 | 2 | 8 | 5 | 0.001 | Accuracy test |
| Wide interval | -2 | 5 | 20 | 30 | 0.05 | Planning n |
Understanding Simpson Rule Error
Why Error Matters
Simpson rule is a strong numerical integration method. It estimates area by fitting parabolas through points. The method is often accurate, but every estimate can still carry error. This calculator helps you measure that risk before trusting a final answer.
Role of the Fourth Derivative
The error bound depends on the fourth derivative. A smooth function with a small fourth derivative usually gives a smaller bound. A function that bends sharply may need more subintervals. The value entered for max |f''''(x)| should be the largest reasonable value on the full interval.
Choosing Subintervals
Composite Simpson rule requires an even number of subintervals. If you enter an odd value, this calculator adjusts it to the next even value. A larger n gives a smaller step size. Since n is raised to the fourth power in the denominator, increasing n can reduce the bound quickly.
Using Tolerance
Tolerance is the largest error you can accept. The calculator compares the estimated bound with your tolerance. It also estimates the minimum even n needed. This is useful when planning tables, assignments, lab reports, and statistical numerical work.
Actual Error Comparison
If you know the exact integral, enter it with the Simpson approximation. The calculator then gives actual error and percent error. This comparison is helpful for checking worked examples. It also shows whether the theoretical bound is conservative.
Good Practice
Use clean inputs and realistic derivative limits. Add a safety factor when the fourth derivative is estimated roughly. Round final answers only after calculation. Keep more internal precision for better reporting. Download the CSV file for spreadsheets. Use the PDF file for a simple saved summary.
Statistics Use
In statistics, integrals appear in probability, density functions, distribution areas, and expectation calculations. Simpson error checks help judge whether a numerical area is reliable. This gives clearer support for conclusions based on approximated probabilities.
FAQs
What does this calculator estimate?
It estimates the theoretical error bound for composite Simpson rule. It can also compare exact and approximate values when both are entered.
Why must n be even?
Simpson rule works with pairs of subintervals. Each pair forms a parabolic estimate. That is why composite Simpson rule needs an even n.
What is max |f''''(x)|?
It is the largest absolute value of the fourth derivative on the interval. The error formula uses this value to form a safe upper bound.
What happens if I enter odd n?
The calculator automatically uses the next even number. It still shows your original n, so you can see the adjustment clearly.
Can this calculate the Simpson approximation itself?
This page focuses on error analysis. You may enter a Simpson approximation if you already calculated it elsewhere.
What is tolerance used for?
Tolerance is your acceptable error limit. The calculator checks whether the error bound is below that limit.
Why add a safety factor?
A safety factor increases the derivative maximum. It is useful when your fourth derivative estimate is uncertain or rounded.
Are CSV and PDF downloads included?
Yes. Submit the form with either download button. The page creates a report using your current input values.