Riemannkvoten

Riemannkvoten is a term that refers to a specific type of sum used in calculus, particularly in the definition of the definite integral. It is named after the German mathematician Bernhard Riemann. The Riemannkvoten is constructed by dividing a given interval on the x-axis into a number of subintervals. Within each subinterval, a point is chosen, and the function's value at that point is multiplied by the width of the subinterval. These products are then summed up.

Mathematically, for a function f(x) over an interval [a, b], a Riemannkvoten is represented as the sum of f(c_i) * Δx_i, where c_i is a point in the i-th subinterval and Δx_i is the width of that subinterval. The accuracy of the Riemannkvoten as an approximation of the area under the curve increases as the number of subintervals increases and the width of each subinterval decreases.

The concept of the Riemannkvoten is fundamental to understanding how definite integrals are defined and calculated. The definite integral of a function over an interval is formally defined as the limit of the Riemannkvoten as the width of the largest subinterval approaches zero. This limit, if it exists, represents the exact area under the curve of the function between the specified limits of integration. Different choices for the point c_i within each subinterval (e.g., left endpoint, right endpoint, midpoint) lead to different specific types of Riemann sums, but they all converge to the same definite integral value.