Midpoint Rule Calculator
Estimate the value of a definite integral by summing the areas of rectangles whose heights are taken at the midpoint of each subinterval.
Function f(x)
Midpoint rule estimate
- 1
Subinterval width Δx = (b − a) ÷ n
(2 − 0) ÷ 4 = 0.5 - 2
Sum of 4 midpoint heights Σf(mᵢ)
5.25 - 3
Integral ≈ Δx × Σf(mᵢ)
0.5 × 5.25 = 2.6250The midpoint rule evaluates the function at the centre of each strip, balancing over- and under-estimates.
How does this calculator work?
The midpoint rule approximates a definite integral by dividing [a, b] into n equal strips of width Δx = (b − a)/n, evaluating the function at each strip's midpoint, and summing those heights times Δx. It is exact for linear functions and its error falls off like 1/n², so larger n gives sharper estimates.
Formula
How this is calculated
The midpoint rule is a Riemann-sum method for numerical integration. You choose a function f(x), the lower limit a, the upper limit b, and the number of equal subintervals n. The calculator first computes the common width Δx = (b − a) / n, splitting [a, b] into n strips.
For each strip i (from 0 to n − 1) it evaluates the function at the midpoint mₚ = a + (i + 0.5)·Δx and treats f(mₚ) as the rectangle height. Multiplying each height by Δx gives a rectangle area, and adding all rectangles yields the approximation Δx · Σ f(mₚ). Using the midpoint rather than a left or right endpoint typically cancels overshoot and undershoot, so the midpoint rule is exact for linear functions and second-order accurate overall.
Limits may be given in either order, but the integral keeps the sign convention of a → b. The error shrinks roughly as 1/n² as you raise n, while functions with sharp curvature or singularities (for example 1/x crossing zero) need more subintervals or a domain that avoids the singular point. The angle-based options (sin, cos) use radians.
Frequently asked questions
Sampling at the midpoint balances the parts of the rectangle that are above and below the curve, so the midpoint rule is more accurate than the left or right Riemann sum for the same n and is exact for straight lines.
Increase the number of subintervals n. The midpoint-rule error decreases approximately in proportion to 1/n², so doubling n cuts the error by about a factor of four.
Yes. If a is greater than b the width Δx becomes negative and the approximation carries the correct sign, matching ∫ₐᵇ = −∫_bᵃ.
Also known as
TG we-Calculate Editorial Team. (2026). Midpoint Rule Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/midpoint-rule-calculator
TG we-Calculate Editorial Team. "Midpoint Rule Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/midpoint-rule-calculator.
TG we-Calculate Editorial Team, "Midpoint Rule Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/midpoint-rule-calculator
@misc{wecalculate_midpoint_rule_calculator, title = {Midpoint Rule Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/midpoint-rule-calculator}}, year = {2026}, note = {TG we-Calculate} }
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