integraal

Integraal is the Dutch term for the mathematical concept known in English as an integral. It is a tool for measuring accumulation, such as area under a curve, total quantity over an interval, or other aggregated values derived from a function.

There are two main senses of an integraal: the indefinite integral and the definite integral. The indefinite integral of a function f is a family of antiderivatives, written as ∫ f(x) dx, whose derivative is f(x). This represents the general solution to the problem of finding a function whose rate of change matches f. The definite integral, written as ∫_a^b f(x) dx, yields a single number representing the accumulated quantity from a to b; for nonnegative f it can be interpreted as the area under the curve f between a and b.

Notation and basic properties are central to the concept. The integral sign ∫, the variable of integration, and the differential dx indicate the variable along which accumulation occurs. Key properties include linearity (the integral of a sum equals the sum of integrals, and constants can be pulled out), and techniques such as substitution (change of variables) and integration by parts. Common methods also involve partial fractions, trigonometric substitutions, and recognizing standard antiderivatives. Not all functions are integrable in the traditional sense; the Riemann integral applies to a broad class of functions, while the Lebesgue integral generalizes integration to a wider set of functions. Improper integrals extend the theory to unbounded intervals or integrands with singularities.

Applications of integraal calculus are widespread, including calculating areas, volumes, arc length, and work in physics, as well as problems in probability and statistics. The fundamental theorem of calculus links differentiation and integration, showing that they are inverse processes: under suitable conditions, the accumulation of f over an interval equals the antiderivative of f evaluated at the endpoints.

Historically, the concept emerged in the 17th century through the work of Newton and Leibniz, with roots in classical geometry and methods for finding areas and volumes.