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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. As n → ∞, the Riemann sum converges to the exact integral. Riemann sums are the conceptual foundation for understanding what integration means geometrically.

Công thức

Divide [a, b] into n equal subintervals of width Δx = (b−a)/n

Hướng dẫn từng bước

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

Ví dụ có lời giải

đầu vào
∫[0,1] x² dx, n=4, midpoint
Kết quả
≈ 0.328125 (exact: 0.3333)
Error decreases as 1/n²
đầu vào
∫[0,1] x² dx, n=100, midpoint
Kết quả
≈ 0.333325
Approaches exact value as n increases

Câu hỏi thường gặp

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.

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