Integraalne

Integraalne is an Estonian adjective meaning "integral" or "related to an integral." In mathematical contexts, integraalne describes concepts and objects derived from the integral, a fundamental tool used to measure accumulation, such as area, mass, or total quantity.

The integral itself comes in two main forms. The indefinite integral, or antiderivative, of a function f is a function F whose derivative is f; it is written as ∫ f(x) dx and represents a family of functions differing by a constant. The definite integral computes a numeric value over an interval [a, b], written as ∫_a^b f(x) dx, and by the Fundamental Theorem of Calculus equals F(b) − F(a) when F' = f.

Generalizations and variants include the Riemann integral, defined via upper and lower sums, and the Lebesgue integral, which extends integration to a broader class of functions with respect to a measure. Other important types are improper integrals, where the interval is unbounded or the integrand has infinite discontinuities, and various context-specific integrals in vector calculus and complex analysis.

Numerical integration approximates integrals when antiderivatives are unknown or difficult to express. Common methods include the trapezoidal rule, Simpson’s rule, and Gaussian quadrature, often used in engineering and scientific computing.

Applications of integraalne concepts appear across physics, engineering, economics, and probability. They provide tools to compute work, probabilities (via expected values), total mass, and cumulative quantities over continuous domains. The term integral traces back to Latin roots meaning “whole” or “unbroken,” reflecting its historical role in summing infinitesimal parts into a whole.