Calculator
Formula Used
The midpoint rule error bound is:
|EM| ≤ K(b - a)3 / (24n2)
Here, K is an upper bound for |f''(x)| on the interval [a, b]. The value n is the number of equal subintervals. The width h is |b - a| / n.
When a tolerance T is supplied, the required panel count is:
n ≥ √(K(b - a)3 / (24T))
If a midpoint sum is entered, the midpoint approximation is:
Mn = h × Σf(midpoints)
How to Use This Calculator
- Enter the lower limit and upper limit of the integral.
- Enter the number of midpoint panels.
- Enter K, the maximum absolute second derivative value.
- Enter a target tolerance if you want a required panel count.
- Add the midpoint sum if you want the numerical midpoint estimate.
- Add the exact integral only when it is known.
- Press Calculate to view the result above the form.
- Use the CSV or PDF button to save the report.
Example Data Table
| a | b | K | n | Tolerance | Error Bound | Required n |
|---|---|---|---|---|---|---|
| 0 | 1 | 2 | 10 | 0.001 | 0.000833 | 10 |
| 1 | 5 | 0.75 | 20 | 0.01 | 0.005000 | 15 |
| 0 | 2 | 3 | 8 | 0.005 | 0.015625 | 15 |
| 2 | 6 | 1.2 | 30 | 0.002 | 0.003556 | 40 |
Midpoint Rule Error Bound in Statistics
Why the Bound Matters
The midpoint rule estimates a definite integral by using rectangles whose heights come from the middle of each subinterval. It often gives strong results because every sample point sits away from the endpoints. Still, every estimate needs a bound. The bound shows the largest expected error before the exact integral is known.
Key Inputs
The error bound depends on four values. The first is the lower limit. The second is the upper limit. The third is the number of equal panels. The fourth is K, a safe upper limit for the absolute second derivative on the whole interval. A larger K means more curvature. More curvature can increase possible error. More panels reduce the error quickly.
Planning Accuracy
This calculator helps you test both directions. You can enter a panel count and get the maximum error. You can also enter a tolerance and see the minimum panel count needed. That is useful for statistics, probability density integrals, expected value checks, and numerical summaries where a controlled approximation is required.
Optional Estimate Check
The optional midpoint sum field adds a practical layer. If you already sampled the function at each midpoint, enter the sum of those function values. The tool multiplies that sum by the panel width. It then returns the midpoint approximation. If you also know the exact integral, the calculator compares the actual error with the theoretical limit.
Choosing K
Use a conservative K when you are unsure. The value should cover the largest possible absolute value of the second derivative across the interval. If K is too small, the stated bound may be unsafe. If K is too large, the bound may be wider than needed, but it remains cautious.
Better Numerical Decisions
The midpoint error formula is useful because it links accuracy to panel count. Doubling the panel count divides the bound by four. This square relationship makes planning easy. You can decide whether a tighter tolerance is worth the extra evaluations. The download buttons help save the result for reports, worksheets, audits, or later review. For classroom use, the calculator also displays an example table. The table shows how interval width, panel count, and curvature affect the final bound. These examples make the formula easier to check before entering custom data. They also reduce simple data entry mistakes.
FAQs
What is the midpoint rule error bound?
It is a theoretical limit for the maximum error in a midpoint rule approximation. It uses interval length, panel count, and a bound for the second derivative.
What does K mean?
K is an upper bound for the absolute value of the second derivative on the full interval. It measures the largest expected curvature.
Why does the second derivative matter?
The midpoint rule error depends on curvature. A function with stronger curvature can produce larger approximation error, so the bound uses |f''(x)|.
What happens when n increases?
The error bound decreases with n squared. If you double n, the theoretical error bound becomes one fourth as large.
Can this calculator find required panels?
Yes. Enter a positive tolerance. The calculator solves the midpoint error formula for n and rounds up to the next whole panel count.
Is the actual error always shown?
No. Actual error is shown only when you enter both the midpoint sum and the exact integral. Otherwise, the calculator shows the theoretical bound.
Can K be zero?
Yes. K can be zero when the second derivative is zero across the interval. In that case, the midpoint rule has a zero theoretical error bound.
Why use CSV and PDF exports?
CSV is useful for spreadsheets and further analysis. PDF is useful for reports, classroom notes, audit records, or saved calculation summaries.