Numerical Integrals

Numerical integration (quadrature) approximates definite integrals ∫ₐᵇ f(x) dx when analytical integration is impossible, impractical, or when f(x) is known only at discrete points (experimental data, tabulated values). The integral represents the area under the curve y = f(x) from x = a to x = b. The trapezoidal rule approximates f(x) as straight-line segments between points. For a single interval: ∫ₐᵇ f(x) dx ≈ (b − a)[f(a) + f(b)] / 2. For n equal subintervals with step size h = (b − a)/n, the composite trapezoidal rule is: ∫ₐᵇ f(x) dx ≈ (h/2)[f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(x_{n−1}) + f(x_n)], where x₀ = a and x_n = b. The error is O(h²), meaning halving h reduces error by approximately a factor of 4. Simpson's 1/3 rule uses quadratic (parabolic) approximation through three points, requiring an even number of subintervals. For n subintervals (n must be even): ∫ₐᵇ f(x) dx ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + 2f(x₄) + ... + 4f(x_{n−1}) + f(x_n)]. The pattern of coefficients is 1, 4, 2, 4, 2, ..., 4, 1. Simpson's 1/3 rule has error O(h⁴), making it much more accurate than trapezoidal for smooth functions. Simpson's 3/8 rule uses cubic approximation through four points and requires n to be a multiple of 3: ∫ₐᵇ f(x) dx ≈ (3h/8)[f(x₀) + 3f(x₁) + 3f(x₂) + 2f(x₃) + 3f(x₄) + 3f(x₅) + 2f(x₆) + ... + f(x_n)]. The coefficient pattern is 1, 3, 3, 2, 3, 3, 2, ..., 1. Error is also O(h⁴). Romberg integration applies Richardson extrapolation to trapezoidal estimates with different step sizes, systematically eliminating error terms to achieve higher accuracy without additional function evaluations. Gaussian quadrature selects optimal evaluation points (not equally spaced) and weights to integrate polynomials of degree 2n−1 exactly using only n points, achieving high accuracy with fewer function evaluations for smooth integrands.