Matematika

Kalkulačka Riemannova součtu

ΣRiemann Sum Calculator

f(x)

n (rectangles)

What is Riemann Sum Calculator?

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Riemannův součet je konečnou aproximací určitého integrálu. Rozdělí interval [a, b] na n podintervalů a aproximuje plochu pod křivkou pomocí n obdélníků. Jako n → ∞ konverguje Riemannův součet k přesnému integrálu. Riemannovy součty jsou koncepčním základem pro pochopení toho, co integrace znamená geometricky.

Calkulon makes complex calculations simple — built for students and everyday problem-solvers.

Vzorec

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f(x)Rozdělte [a, b] na n stejných podintervalů o šířce Δx = (b−a)/n

How to Riemann Sum Calculator

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1Divide [a, b] into n equal subintervals of width Δx = (b−a)/n
2Left Riemann: use left endpoint of each subinterval — Σ f(xᵢ) · Δx
3Right Riemann: use right endpoint — Σ f(xᵢ₊₁) · Δx
4Midpoint Riemann: use midpoint — Σ f((xᵢ+xᵢ₊₁)/2) · Δx
5Midpoint rule is generally more accurate than left or right for the same n

Worked Examples

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Example 1

Given:∫[0,1] x² dx, n=4, midpoint

Výsledek:≈ 0.328125 (exact: 0.3333)

Error decreases as 1/n²

Example 2

Given:∫[0,1] x² dx, n=100, midpoint

Výsledek:≈ 0.333325

Approaches exact value as n increases

Real-World Applications

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Academic homework and exam preparation

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Engineering and architecture calculations

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Everyday measurement tasks around the home

Riemann Sum Method Comparison

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Method	Formula	Error Order	Best For
Left	Σ f(xᵢ) · Δx	O(Δx)	Decreasing functions
Right	Σ f(xᵢ₊₁) · Δx	O(Δx)	Increasing functions
Midpoint	Σ f(mid) · Δx	O(Δx²)	General use
Trapezoid	(Δx/2)[f(a) + 2Σf(xᵢ) + f(b)]	O(Δx²)	Smooth functions
Simpson	(Δx/3)[f(a)+4f(x₁)+2f(x₂)+...+f(b)]	O(Δx⁴)	Smooth functions, best accuracy

Frequently Asked Questions

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What is Riemann Sum Is A Finite Approximation Of A Definite Integral?

A Riemann sum is a finite approximation of a definite integral. It partitions the interval [a, b] into n subintervals and approximates the area under the curve using n rectangles

How accurate is the Riemann Sum Is A Finite Approximation Of A Definite Integral calculator?

The calculator uses the standard published formula for riemann sum is a finite approximation of a definite integral. Results are accurate to the precision of the inputs you provide. For financial, medical, or legal decisions, always verify with a qualified professional.

What units does the Riemann Sum Is A Finite Approximation Of A Definite Integral calculator use?

This calculator works with inches. You can enter values in the units shown — the calculator handles all conversions internally.

What formula does the Riemann Sum Is A Finite Approximation Of A Definite Integral calculator use?

The core formula is: Divide [a, b] into n equal subintervals of width Δx = (b−a)/n. Each step in the calculation is shown so you can verify the result manually.

References

Deep Dive

Read the full guide on how to use this calculator effectively

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Reviewed October 2026

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