∫

Riemann Integral

∫ f(x) dx = lim(N→∞) Σ f(xᵢ)·Δx — the area under a curve as the limit of rectangle sums

Mathematics Calculus Numerical Methods Convergence

Function:

Method:

Subintervals N 10 a 0.0 b 3.14

N = 10 Riemann sum ≈ — Exact ∫ = — Error = — Δx = —

∫ The Riemann Integral

The definite integral is defined as the limit of Riemann sums: ∫ₐᵇ f(x) dx = lim(N→∞) Σᵢ f(x*ᵢ)·Δx where Δx = (b−a)/N and x*ᵢ is a sample point in the i-th subinterval.

Four classic approximation rules:

Left: x*ᵢ = a + i·Δx — uses left endpoint of each rectangle. Error O(Δx).
Right: x*ᵢ = a + (i+1)·Δx — uses right endpoint. Also O(Δx).
Midpoint: x*ᵢ = a + (i+½)·Δx — more accurate, error O(Δx²).
Trapezoid: ½(f(xᵢ)+f(xᵢ₊₁))·Δx — connects top corners with a line. Error O(Δx²).

The error bound: |Error| ≤ M·(b−a)³/(12N²) for Midpoint and Trapezoid (M = max|f″|). Midpoint is exactly twice as accurate as Trapezoid for smooth functions, and together form Simpson's Rule = (2×Midpoint + Trapezoid) / 3 with error O(Δx⁴).