Calculator
Example Data Table
| Function | Lower Limit | Upper Limit | Method | Intervals | Approximate Result |
|---|---|---|---|---|---|
| x^2 | 0 | 3 | Simpson's 1/3 Rule | 12 | 9.000000 |
| sin(x) | 0 | pi | Simpson's 1/3 Rule | 12 | 2.000053 |
| exp(-x^2) | -1 | 1 | Midpoint Rule | 20 | 1.494062 |
Formula Used
Definite Integral
The target quantity is the signed area under the curve:
∫ab f(x) dx
Trapezoidal Rule
∫ab f(x) dx ≈ h [ (f(a)+f(b))/2 + Σ f(a+ih) ]
Midpoint Rule
∫ab f(x) dx ≈ h Σ f(a + (i+1/2)h)
Simpson's 1/3 Rule
∫ab f(x) dx ≈ (h/3)[f(x0) + f(xn) + 4Σf(xodd) + 2Σf(xeven)]
Here, h = (b - a) / n. Larger n values often improve the estimate, especially for curved functions.
How to Use This Calculator
- Enter the function using x as the variable.
- Provide the lower and upper integration limits.
- Choose the number of subintervals for the numerical estimate.
- Select Simpson, trapezoidal, or midpoint as the primary method.
- Set graph points and decimal places for display control.
- Press the calculate button to show the result above the form.
- Review the comparison table, graph, and sampled values.
- Use the export buttons to save summary results as CSV or PDF.
FAQs
1) What does this calculator compute?
It estimates a definite integral between two limits. The result represents signed area, so sections below the x-axis subtract from sections above it.
2) Which method is usually most accurate?
Simpson's rule often gives better accuracy for smooth curves. Trapezoidal and midpoint rules remain useful for comparison and quick checks.
3) Why can Simpson change my interval count?
Simpson's 1/3 rule requires an even number of subintervals. If you enter an odd value, the calculator increases it by one automatically.
4) Can I use pi and common functions?
Yes. You can use functions like sin(x), cos(x), sqrt(x), abs(x), log(x), exp(x), and the constant pi.
5) What if the lower limit is greater than the upper limit?
The calculator still works. The step size becomes negative, so the signed integral changes sign as expected.
6) What is the absolute area estimate?
Absolute area ignores sign by integrating |f(x)| across the same interval. It is useful when the curve crosses the x-axis.
7) Why might the calculator show an error?
Errors appear when the function is undefined on part of the interval, contains unsupported symbols, or produces invalid numeric values.
8) Does a larger interval count always help?
Usually yes, but not always dramatically. Higher counts improve resolution, yet very difficult functions may still need careful interpretation.