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Midpoint Riemann Sum — result sheet
Estimates a definite integral from function values sampled at the midpoint of each equal subinterval.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: Delta x = (b-a)/n; integral approx Delta x x sum of f(midpoint_i).
Divides the interval into equal pieces and samples the function at each piece's midpoint. Multiplying the average rectangle width by the sum of midpoint heights gives a useful integral estimate without requiring the function formula.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Lower endpoint a — minimum -1000000000000; maximum 1000000000000
- Upper endpoint b — Must be greater than a.; minimum -1000000000000; maximum 1000000000000
- Midpoint heights — One f-value at each subinterval midpoint; enter 1-1,000 comma- or space-separated values.
Worked example
| Input | Value |
|---|---|
| Lower endpoint a | 0 |
| Upper endpoint b | 2 |
| Midpoint heights | 1, 3 |
Integral estimate 4.000000 with width 1.000000 across 2 intervals.
Assumptions and limits
- The entered heights are function values at the midpoint of each equal subinterval.
- The function is reasonably smooth on the interval; more subintervals generally reduce approximation error.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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