arealfunktionen

Arealfunktionen is a term used in mathematics to describe how the area of a region depends on a parameter, typically the upper limit of integration. In calculus, the arealfunktionen A is commonly defined for a function f on an interval [a, b] by A(x) = ∫_a^x f(t) dt for x in [a, b]. This function measures the area between the graph of f and the horizontal axis as x progresses from a to a given value, and it can represent signed area if f changes sign.

Properties: If f is continuous on [a, b], then A is differentiable on (a, b) and A'(x) = f(x) by the Fundamental Theorem of Calculus. A(a) = 0. A is nondecreasing wherever f ≥ 0 and nonincreasing wherever f ≤ 0. If f is integrable, A is continuous as well.

Examples: For f(x) = x^2 on [0, x], A(x) = ∫_0^x t^2 dt = x^3/3. As a geometric example, the area of a circle as a function of its radius r is A(r) = π r^2.

Applications: Arealfunktionen is used to compute areas of regions bounded by curves and to understand how area accumulates as a parameter changes. In probability, the cumulative distribution function is the area under a density function up to a given point. The concept can be extended to multivariable contexts, where double integrals describe volumes instead of areas.