integraler

Integraler are a central concept in calculus that quantify accumulation. In the most common sense, an integral assigns a number to a function over an interval, capturing quantities such as area, total change, or accumulated resources. There are two main notions: the indefinite integral (an antiderivative F of a function f, with F′(x) = f(x)) and the definite integral, the limit of Riemann sums and written as ∫_a^b f(x) dx. The definite integral often represents the accumulated amount between a and b.

The Riemann integral provides a rigorous framework for when a function is integrable on a closed interval. If a function is bounded and its upper and lower sums converge to the same limit, the function is integrable and the limit is the definite integral. More general formulations exist, notably the Lebesgue integral, which extends integration to a wider class of functions and is central to measure theory and probability.

Key properties of integrals include linearity (the integral of a linear combination is the corresponding combination of integrals) and additivity over intervals. Common techniques used to evaluate integrals include substitution, integration by parts, partial fractions, and trigonometric substitutions. The Fundamental Theorem of Calculus links differentiation and integration: under suitable conditions, differentiation and integration are inverse processes, with the integral of f over [a,b] equaling F(b) − F(a) when F is an antiderivative of f.

Applications of integrals span physics, engineering, statistics, economics, and beyond, including calculations of work, areas, probabilities, and expected values. The concept has historical roots in the work of Newton and Leibniz in the 17th century and was later formalized through developments in analysis.