Calculator Inputs
The calculator uses one-column page flow, while the input fields shift to three columns on large screens, two on tablets, and one on mobile.
Example data table
| Function | Interval | Samples | Replications | Estimated Integral | Exact Value | Absolute Error |
|---|---|---|---|---|---|---|
| x² | [0, 1] | 10,000 | 5 | 0.333921 | 0.333333 | 0.000588 |
| sin(x) | [0, π] | 50,000 | 3 | 1.999412 | 2.000000 | 0.000588 |
| e^x | [0, 1] | 25,000 | 4 | 1.719103 | 1.718282 | 0.000821 |
| 1 / (1 + x²) | [0, 1] | 30,000 | 4 | 0.785644 | 0.785398 | 0.000246 |
Formula used
Monte Carlo integration estimates a definite integral by averaging function values at randomly sampled points from the interval.
For a basic estimator, the points xi are sampled uniformly from the interval [a, b].
Here, s is the sample standard deviation of the evaluated function values.
When stratified sampling is enabled, the interval is split into smaller parts, and each part contributes its own weighted average.
In that expression, hj is the width of stratum j, and f̄j is the sample mean inside that stratum.
How to use this calculator
- Select a function family or choose the polynomial option.
- Enter the lower and upper bounds of integration.
- Set the number of samples for each replication.
- Choose how many replications you want for stability checks.
- Select a confidence level for interval reporting.
- Optionally add a random seed for reproducible runs.
- Enable stratified sampling or antithetic pairs if needed.
- Enter a known exact value to compare estimation error.
- Press the calculate button to display results above the form.
- Download the generated CSV or PDF report if required.
FAQs
1. What does Monte Carlo integration do?
It estimates a definite integral using random samples from the chosen interval. The method is useful when exact antiderivatives are difficult or when high-dimensional extensions are needed.
2. Why do results change between runs?
The estimator depends on random draws, so each run can differ slightly. Adding more samples usually reduces variation. A fixed seed makes repeated runs reproducible.
3. What is a good sample size?
Larger samples generally improve precision. Start with 10,000 or more for smooth one-dimensional functions, then increase until the standard error and confidence interval are acceptably small.
4. What does the confidence interval mean?
It gives a range around the estimate based on sampling variability. Wider intervals suggest more uncertainty, while narrower intervals suggest better precision for the selected settings.
5. When should I use stratified sampling?
Use it when you want steadier coverage across the interval. It often lowers variance because each subrange contributes directly instead of relying on uneven random clustering.
6. What are antithetic pairs?
Antithetic pairing samples mirrored random points. This can reduce variance for many smooth functions by balancing high and low function values more effectively than basic random sampling alone.
7. Why enter a known exact value?
An exact benchmark lets you measure absolute and relative error. That makes it easier to compare different sample sizes, methods, and variance-reduction settings objectively.
8. Can this handle custom functions?
This version includes several common families and a polynomial option. For many classroom and analysis tasks, that covers standard demonstrations and method comparisons well.