Integralelor

Integralelor are a central concept in calculus, referring to the mathematical notion of integrals in their plural form. They describe accumulation and, in geometric terms, the area under a curve. There are two main families: indefinite integrals, also known as antiderivatives, and definite integrals.

An indefinite integral of a function f is a function F whose derivative is f; it is written as ∫ f(x) dx = F(x) + C, where C is an arbitrary constant. A definite integral, written as ∫_a^b f(x) dx, yields a number that represents, for example, the net area between the graph of f and the x-axis on the interval [a, b], or the total accumulation of a quantity described by f over that interval.

The Fundamental Theorem of Calculus links differentiation and integration: if f is continuous on [a, b] and F is an antiderivative of f, then ∫_a^b f(x) dx = F(b) − F(a). This theorem shows that integration is the inverse operation of differentiation in a precise sense.

Types and generalizations of integralelor include Riemann integrals, which capture the classical notion of area, and Lebesgue integrals, which extend integration to a broader class of functions. Improper integrals extend the idea to unbounded intervals or unbounded integrands. In higher dimensions, integralelor generalize to multiple integrals and to line, surface, and contour integrals, used in physics, engineering, and complex analysis.

Computing integralelor relies on techniques such as substitution, integration by parts, partial fractions, and numerical methods like the trapezoidal and Simpson rules. Applications span geometry, probability, physics, economics, and beyond, making integralelor a foundational tool in mathematical analysis.