Advanced Simpson’s 1/3 Rule Calculator
Example Data Table
This example estimates the area under f(x) = x² + 1 from 0 to 4.
| Index | x | f(x) | Weight | Weighted Value |
|---|---|---|---|---|
| 0 | 0 | 1 | 1 | 1 |
| 1 | 1 | 2 | 4 | 8 |
| 2 | 2 | 5 | 2 | 10 |
| 3 | 3 | 10 | 4 | 40 |
| 4 | 4 | 17 | 1 | 17 |
Result: h = 1, weighted sum = 76, integral = 25.333333.
Formula Used
Simpson’s 1/3 rule approximates a definite integral by fitting parabolic arcs through pairs of subintervals.
Integral ≈ h / 3 × [y₀ + yₙ + 4(y₁ + y₃ + ... + yₙ₋₁) + 2(y₂ + y₄ + ... + yₙ₋₂)]
Here, h = (b - a) / n. The value n must be even. The x values must be equally spaced when table input is used.
How to Use This Calculator
Select tabulated data mode when you already have x and y values. Enter values separated by commas, spaces, or semicolons.
Select function mode when you know the expression. Enter the lower limit, upper limit, and an even number of subintervals.
Press the calculate button. The result will appear above the form. Review the weighted table, then export the result as CSV or PDF.
Understanding Simpson’s 1/3 Rule
What the Method Does
Simpson’s 1/3 rule is a numerical integration method. It estimates the area under a curve. The method is useful when exact integration is hard. It also helps when only measured data points are available. Instead of using straight lines, it uses parabolic segments. This usually gives better accuracy than the trapezoidal rule.
Why Equal Spacing Matters
The method depends on a constant step size. This step size is called h. Each x value must be separated by the same interval. If spacing changes, the standard formula becomes invalid. This calculator checks spacing for table data. It also checks that the number of subintervals is even.
How Weights Work
The first and last y values receive weight one. Odd interior values receive weight four. Even interior values receive weight two. These weights create the parabolic estimate. The weighted sum is then multiplied by h divided by three. The result is the estimated definite integral.
Advanced Uses
This calculator supports two workflows. You can enter measured x and y values. You can also enter a function with limits. Function mode creates points automatically. It then applies the same Simpson pattern. This is helpful for engineering, physics, economics, and statistics tasks.
Accuracy Notes
Accuracy improves when the function is smooth. More subintervals often improve the estimate. Very sharp curves may need smaller spacing. Discontinuous functions should be handled carefully. Always compare results when precision is important.
Frequently Asked Questions
1. What is Simpson’s 1/3 rule?
It is a numerical integration method. It estimates the area under a curve using parabolic segments instead of straight lines.
2. Why must subintervals be even?
The rule works on pairs of subintervals. Each pair forms one parabolic segment. Therefore, the total number must be even.
3. Can I use unequal x values?
No. The standard Simpson’s 1/3 rule requires equal spacing between x values. Use another method for unequal spacing.
4. What does h mean?
The value h is the step size. It is the distance between consecutive x values or calculated from limits and subintervals.
5. Which weights are used?
The first and last values use weight one. Odd interior values use four. Even interior values use two.
6. Is this better than the trapezoidal rule?
Usually yes for smooth curves. Simpson’s rule uses parabolas, so it often gives a more accurate estimate.
7. Can I export my result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple report summary.
8. What functions are supported?
Function mode supports common operations and terms such as powers, sin, cos, tan, sqrt, log, exp, and abs.