Midpoint Rule Calculator
Work out midpoint rule instantly with clear inputs, formula shown and shareable results.
About Midpoint Rule Calculator
The midpoint rule uses the function value at the centre of each strip, which cancels the linear part of the error. It is typically about twice as accurate as the trapezoid rule and errs in the opposite direction.
Formula
Midpoint rule
∫ ≈ h·Σ f(a + (k + ½)h)
Worked example
With the default values below, Midpoint Rule Calculator returns Midpoint rule estimate of 8.6481481481. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Polynomial coefficients, highest power first
- 1, 0, 0
- From x =
- 1
- To x =
- 3
- Number of intervalsaccepts 1–10,000
- 6
Results
- Midpoint rule estimate
- 8.6481481481
- Exact integral
- 8.6666666667
- Absolute error
- 0.018518518519
- Trapezoid rule for comparison
- 8.7037037037
Step by step
- 1Values used
- 2Midpoint rule
- 3Midpoint rule estimate= 8.6481481481
- 4Exact integral= 8.6666666667
- 5Absolute error= 0.018518518519
- 6Trapezoid rule for comparison= 8.7037037037
Assumptions and limits
- The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 12 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
Integrate x² from 0 to 3 with 6 intervals?
The exact value is 9 and the midpoint rule gives 8.9375, an error of 0.0625.
Why is it better than the trapezoid rule?
The trapezoid rule's error is about twice as large and of opposite sign; averaging the two in a 2:1 ratio gives Simpson's rule.