Trapezoidal rule

The trapezoidal rule approximates a definite integral by summing the areas of trapezoids between adjacent sample points. It is exact for linear functions and generally has second-order error.

Numerical integration: Numerical integration estimates definite integrals using finite computations, often from sampled function values. The trapezoidal rule is one basic method for approximating definite integrals.
Definite integral: A definite integral gives the signed accumulation of a function over an interval. The trapezoidal rule estimates this accumulated quantity.
Numerical solution of differential equations: Numerical methods approximate solutions of differential equations when exact solutions are unavailable or impractical. Integral forms of differential equations can be evaluated with trapezoidal quadrature.
Simpson’s rule: Simpson’s rule approximates integrals by integrating quadratic interpolants over pairs of subintervals. It usually has higher order accuracy for smooth functions but requires paired intervals.
Composite trapezoidal rule: The composite trapezoidal rule applies the trapezoidal rule across multiple subintervals and sums their areas. It extends a single trapezoid to many adjacent sample intervals.
Linear interpolation: Linear interpolation estimates values between known points by joining them with a straight line. Each trapezoid corresponds to integrating the line between two sampled values.
Signal processing: Signal processing analyzes, transforms, or synthesizes signals such as audio, images, and measurements. The rule estimates signal energy and accumulated quantities from discrete samples.
Gaussian quadrature: Gaussian quadrature chooses sample nodes and weights to integrate polynomials of high degree exactly. It can achieve high accuracy with fewer function evaluations when arbitrary nodes are available.
Quadrature error: Quadrature error is the difference between a numerical approximation to an integral and its exact value. Its leading error term explains why halving the interval width typically reduces error by about fourfold.
Trapezoid: A trapezoid is a quadrilateral with at least one pair of parallel sides. The rule computes each strip's area using this shape.

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