Integraalit

Integraalit, or the integral, is a central concept in calculus that expresses accumulation and serves as the inverse operation to differentiation. There are two main senses: an indefinite integral, which represents the family of antiderivatives of a function, and a definite integral, which assigns a numerical value to the accumulation over a specific interval.

Notation and interpretation are essential. The indefinite integral is written as ∫ f(x) dx and corresponds to finding a function F whose derivative is f, i.e., F' = f. The definite integral is written as ∫_a^b f(x) dx and yields the total accumulation of f over [a, b], such as area, length, or physical quantities that sum along a path. Geometrically, for nonnegative f, the definite integral equals the area under the curve y = f(x) from a to b; more generally, it accounts for signed accumulation.

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 underpins many techniques of evaluation, such as substitution and integration by parts. Practical methods include analytical techniques for functions with closed forms and numerical methods—trapezoidal, Simpson’s rule—for cases lacking elementary antiderivatives.

Applications of integraalit span physics (work and energy), probability (expected values and distributions), engineering, and economics. The concept has roots in ancient geometry and was formalized in the 17th century by Newton and Leibniz, with the integral sign ∫ introduced by Leibniz.