Riemannintegrasjon

Riemannintegrasjon, also known as Riemann integration, is a fundamental concept in calculus used to define the definite integral of a function. It provides a way to approximate the area under the curve of a function by dividing the area into a series of small rectangles. The width of each rectangle is determined by a partition of the interval over which the function is being integrated, and the height is determined by the function's value at a chosen point within that subinterval.

The Riemann integral is defined as the limit of the sum of the areas of these approximating rectangles as the width of the widest rectangle approaches zero. This limit, if it exists, is the value of the definite integral. For a function f(x) defined on the closed interval [a, b], the Riemann integral is denoted by the integral symbol ∫ from a to b of f(x) dx.

A function is said to be Riemann integrable if its Riemann integral exists. Continuous functions on a closed interval are always Riemann integrable. The concept of Riemann integration is crucial for many areas of mathematics and physics, including calculating areas, volumes, work, and probability. It forms the basis for more advanced integration theories, such as the Lebesgue integral.