integrators

Integrators are devices or algorithms that perform integration, the process of summing infinitesimal contributions to obtain a total over a domain such as time or space. In mathematics, integration provides an antiderivative and a measure of accumulation; in engineering, an electronic or digital integrator accumulates signals.

In calculus, the indefinite integral represents a family of antiderivatives; the definite integral computes a numerical value representing accumulated quantity, such as area or total mass. The fundamental theorem of calculus links differentiation and integration.

Numerical integration, or quadrature, approximates integrals when a function is known only by samples or is difficult to integrate analytically. Common methods include trapezoidal rule, Simpson's rule, and Gaussian quadrature; adaptive quadrature and Monte Carlo methods are used for high-dimensional or irregular domains.

In signal processing and control theory, an ideal integrator has a transfer function 1/s and implements accumulation over time; in discrete time the accumulator sums samples. Real-world integrators face issues such as drift, saturation, quantization, and finite precision.

Applications span physics, statistics, economics, and engineering. Variants include symbolic integrators in computer algebra systems, and numerical integrators for ordinary and partial differential equations, such as algorithms for evaluating volume integrals in finite element methods.