Riemann Sum Calculator

How to Use This Calculator

1. Enter the function f(x) (e.g. x^2, sin(x)).
2. Set lower bound a and upper bound b.
3. Set n (number of rectangles).
4. Choose method: left, right, or midpoint.
5. The Professional tab compares trapezoid, Simpson's, and reference values with error %.

Formula

Example

Frequently Asked Questions

• What is a Riemann sum? ▼
  A Riemann sum approximates the area under a curve by dividing [a,b] into n rectangles and summing f(xᵢ)·Δx. Left, right, and midpoint sums differ in where xᵢ is sampled within each subinterval.
• Which Riemann sum method is most accurate? ▼
  The midpoint rule has O(h²) error (same as trapezoid). Simpson's rule achieves O(h⁴) accuracy by combining trapezoid and midpoint estimates, making it far more accurate for smooth functions.
• What is the trapezoid rule? ▼
  The trapezoid rule uses trapezoids instead of rectangles: T = (Δx/2)[f(x₀) + 2f(x₁) + ... + 2f(xₙ₋₁) + f(xₙ)]. Error is proportional to h² (where h = Δx).
• What is Simpson's rule? ▼
  Simpson's rule is (Δx/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + f(xₙ)] with n even. It fits parabolas through pairs of intervals and achieves O(h⁴) error, exact for polynomials up to degree 3.
• How many rectangles n do I need for a good approximation? ▼
  For midpoint/trapezoid, error ≈ C·h². Doubling n reduces error by 4×. For Simpson's, doubling n reduces error by 16×. For most smooth functions, n=1000 with Simpson gives nearly machine-precision results.

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