Calculator Inputs
Supported functions: sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, sqrt, abs, log, log10, exp, pow, floor, ceil, round, min, max.
Tips: Use ^ for powers, pi for π, e for Euler’s constant, and explicit multiplication such as 2*x.
Example Data Table
This example uses f(x) = x^4 on [0, 2] with n = 4 and the composite 1/3 rule.
| Index | x | f(x) | Weight | Weighted Value |
|---|---|---|---|---|
| 0 | 0.0 | 0.0000 | 1 | 0.0000 |
| 1 | 0.5 | 0.0625 | 4 | 0.2500 |
| 2 | 1.0 | 1.0000 | 2 | 2.0000 |
| 3 | 1.5 | 5.0625 | 4 | 20.2500 |
| 4 | 2.0 | 16.0000 | 1 | 16.0000 |
Here, h = 0.5 and the estimate is (h/3) × 38.5 = 6.4167.
Formula Used
Composite Simpson 1/3: Integral ≈ (h / 3) × [f(x0) + f(xn) + 4 × odd terms + 2 × even terms].
Composite Simpson 3/8: Integral ≈ (3h / 8) × [f(x0) + f(xn) + 3 × most interior terms + 2 × every third interior term].
Automatic mixed option: When n is odd and not a multiple of 3, the page applies 1/3 first and finishes with 3/8 on the final three intervals.
Step size: h = (b - a) / n.
Refined estimate: The page recomputes the integral with 2n subintervals.
Estimated error: |I(2n) - I(n)| / 15. This gives a practical refinement-based accuracy check for smooth functions.
How to Use This Calculator
- Enter the function in terms of x, such as
x^4orsin(x). - Provide the lower and upper integration limits.
- Choose the number of subintervals. Use an even n for 1/3 or a multiple of 3 for 3/8.
- Select a method or keep Auto / Mixed Simpson for flexible handling.
- Set the decimal precision and submit the form.
- Review the approximate integral, error estimate, and row-by-row contribution table.
- Use the CSV button for spreadsheets or the PDF button for a shareable report.
Frequently Asked Questions
1) What does Simpson Rule estimate?
It estimates a definite integral by fitting low-degree polynomials across equally spaced points. This usually gives better accuracy than basic rectangle or trapezoid approaches for smooth functions.
2) When should I use 1/3 or 3/8?
Use 1/3 when n is even. Use 3/8 when n is a multiple of three. The automatic mixed option helps when n is odd but still suitable for combined Simpson handling.
3) Why can’t every n work with one method?
Each Simpson formula groups subintervals in a fixed pattern. The 1/3 rule needs pairs of intervals, while the 3/8 rule needs groups of three.
4) Can I integrate trigonometric or exponential functions?
Yes. The calculator accepts many common functions, including trigonometric, exponential, logarithmic, and root expressions, as long as the function stays finite over the chosen interval.
5) Why is my estimate different from the exact answer?
Numerical integration is approximate. Differences usually shrink when you increase n, especially for smooth functions. Rapid oscillation, singular behavior, or poor spacing can increase error.
6) Does the calculator work when a is greater than b?
Yes. The step size becomes negative, so the resulting integral changes sign. That behavior matches the standard rule for reversing integration limits.
7) What expressions are supported?
Use x, numbers, parentheses, operators, and supported functions like sin, sqrt, log, exp, and pow. Write multiplication clearly, such as 3*x or 2*(x+1).
8) How reliable is the displayed error estimate?
It is a practical refinement check based on comparing n and 2n results. It is useful for smooth functions, but it is not a guaranteed exact bound.