Integroj

Integroj, in mathematical usage, are integrals. In calculus, an integro describes a quantity that accumulates, such as area under a curve, total mass, or probability. There are two main kinds: indefinite integro (antiderivatives) and definite integro. The standard notation uses the integral sign ∫ with respect to a variable, for example ∫ f(x) dx. A definite integro ∫_a^b f(x) dx represents accumulation from a to b, while an indefinite integro ∫ f(x) dx denotes a family of antiderivatives.

The fundamental theorem of calculus links differentiation and integration: the derivative of an antiderivative is the original function, and the definite integro equals the net accumulation over the interval. Evaluation methods include symbolic techniques such as substitution, integration by parts, partial fractions, and recognizing standard forms. When a closed-form antiderivative does not exist, numerical methods approximate values, with the trapezoidal rule, Simpson's rule, and Gaussian quadrature being common choices. In higher dimensions, multiple integro and line or surface integrals extend these ideas, often evaluated via coordinate changes or theorems from vector calculus.

Applications of integroj are widespread: calculating areas and volumes, determining work in physics, computing probabilities and expected values in statistics, and analyzing distributions in economics and engineering. In Esperanto texts, integroj is the plural form used for multiple integrals and related concepts, paralleling its counterparts in other languages.

Historically, the calculus of integrals emerged in the 17th century from the work of Isaac Newton and Gottfried Wilhelm Leibniz, who developed the notation and ideas that underpin modern analysis.