Calculation result
Step-by-step solution
Calculation table
| i | xᵢ | f(xᵢ) | Coefficient | Weighted value | Running sum |
|---|
Graph
Integral inputs
Formula used
Simpson’s rules approximate smooth curves with fitted polynomial segments. They usually outperform simple trapezoids for equal spacing. Accuracy improves when the interval becomes more finely divided.
Simpson’s 1/3 Rule: ∫ₐᵇ f(x)dx ≈ (h/3)[f(x₀) + f(xₙ) + 4Σf(xᵢ odd) + 2Σf(xᵢ even)] Simpson’s 3/8 Rule: ∫ₐᵇ f(x)dx ≈ (3h/8)[f(x₀) + f(xₙ) + 3Σf(xᵢ not divisible by 3) + 2Σf(xᵢ divisible by 3)]
How to use
Enter a function or provide evenly spaced data points. Choose a rule and valid subinterval count. Then calculate, inspect steps, and export your complete results.
- Select function input or tabulated data.
- Enter limits, values, and a valid subinterval count.
- Choose the method, precision, and optional exact value.
- Calculate and review the table, graph, and errors.
Example data
| Function | a | b | n | Suggested method |
|---|---|---|---|---|
| x⁴ − 2x + 1 | 0 | 2 | 8 | Simpson’s 1/3 |
| sin(x) | 0 | π | 12 | Either rule |
| e−x² | 0 | 1 | 12 | Automatic |
Important limitations
Simpson’s rules expect smooth behaviour across the selected interval. Singularities can produce misleading or undefined numerical estimates. Inspect warnings carefully before relying on any reported approximation.
Frequently asked questions
What is Simpson’s Rule?
It estimates a definite integral using polynomial curve segments. The method uses weighted function values at equal intervals. Smooth functions usually produce accurate results with modest sampling.
Why must n be even for the 1/3 Rule?
Each parabolic segment spans two equal subintervals. Therefore, the full interval needs paired subintervals. Odd counts leave one interval without a matching pair.
Why must n divide by three for the 3/8 Rule?
Each cubic grouping spans three equal subintervals. Complete groups require a count divisible by three. Other counts require a compatible mixed-rule calculation for completion.
What does automatic mixed mode do?
It first checks standard method requirements. Otherwise, it combines compatible 1/3 and 3/8 groups. This safely supports many practical subinterval counts during calculations.
Can reversed limits be used?
Yes, reversed limits are accepted automatically. The calculator evaluates the swapped interval internally. It then applies the correct negative sign to results.
Can I enter experimental data?
Yes, use the tabulated data input mode. Values must use equally spaced x-coordinates. The calculator checks equal spacing before applying Simpson weights.
How is percentage error calculated?
Enter a trusted exact integral value first. The calculator compares its approximation against that value. Percentage error uses the exact value as its denominator.
Does increasing n always improve accuracy?
It usually improves accuracy for smooth functions. Floating-point noise can eventually limit further improvement. Discontinuities can still invalidate the expected numerical convergence pattern.
Which angle mode should I choose?
Use radians for standard calculus problems. Choose degrees only when input angles use degrees. The selected mode changes all trigonometric function evaluations accordingly.