Calculator Inputs
Example Data Table
Sample setup: f(x) = 2x² + x, g(x) = 3x + 1, interval [0, 2], midpoint rule, 4 partitions.
| i | xi-1 | xi | ξi | f(ξi) | Δgi | Term |
|---|---|---|---|---|---|---|
| 1 | 0.00 | 0.50 | 0.25 | 0.3750 | 1.5000 | 0.5625 |
| 2 | 0.50 | 1.00 | 0.75 | 1.8750 | 1.5000 | 2.8125 |
| 3 | 1.00 | 1.50 | 1.25 | 4.3750 | 1.5000 | 6.5625 |
| 4 | 1.50 | 2.00 | 1.75 | 7.8750 | 1.5000 | 11.8125 |
Formula Used
Riemann–Stieltjes approximation:
∫ab f(x) dg(x) ≈ Σ f(ξi) [g(xi) − g(xi−1)]
Here, ξi is the sample point chosen from each subinterval. This calculator supports left, right, midpoint, and trapezoid-style sampling.
When g is differentiable, the integral often matches ∫ab f(x)g′(x)dx, which helps interpret the weighting introduced by g.
How to Use This Calculator
- Enter the lower and upper integration bounds.
- Choose the number of partitions for the approximation.
- Select a sampling rule for each partition.
- Define the integrand f(x) with a function type and coefficients.
- Define the integrator g(x) with its own type and coefficients.
- Click the compute button to generate the approximation.
- Review the summary, graph, and detailed partition table.
- Export the visible results as CSV or PDF.
FAQs
1. What does this calculator estimate?
It estimates a Riemann–Stieltjes integral numerically. The tool sums weighted increments of g(x), using function values from chosen sample points inside each partition.
2. Why is this different from an ordinary integral?
An ordinary integral accumulates against dx. A Stieltjes integral accumulates against changes in g(x), so the weighting depends on how the integrator behaves across the interval.
3. Which sampling rule is usually best?
Midpoint often gives stable approximations for smooth functions. Left and right rules help compare directional bias. The trapezoid-style option averages endpoint values for a balanced estimate.
4. What happens if my function is undefined?
The calculator stops and shows an error. This commonly occurs with logarithmic or reciprocal choices when the interval crosses invalid or singular input values.
5. Can I reverse the bounds?
Yes. The calculator normalizes the interval internally and then applies the correct sign change to the final result, preserving orientation properly.
6. Why does the partition count matter?
More partitions usually improve approximation quality, especially when f or g changes rapidly. They also reveal convergence behavior when you compare repeated runs.
7. What does total variation tell me here?
It measures the accumulated absolute change of g across the partition. Large variation means stronger or more oscillatory weighting in the Stieltjes sum.
8. When does this resemble ∫ f(x)g′(x)dx?
When g is differentiable and sufficiently regular, the Stieltjes integral often equals the ordinary integral of f(x) times g′(x), giving a useful interpretation.