Riemannsummák

Riemannsummák is a method for approximating the area under a curve. It involves dividing the region into a series of rectangles and summing their areas. The width of each rectangle is determined by dividing the interval over which the area is being calculated into a specific number of subintervals. The height of each rectangle can be determined in several ways, commonly using the function's value at the left endpoint, right endpoint, or midpoint of each subinterval.

This approximation becomes more accurate as the number of rectangles increases, meaning the width of each rectangle decreases. In the limit, as the number of rectangles approaches infinity, the sum of the areas of these rectangles converges to the exact area under the curve. This limiting process is the foundation of definite integration. The concept was introduced by Bernhard Riemann, a German mathematician, and is fundamental to understanding integral calculus. Different types of Riemann sums, such as left, right, and midpoint sums, provide slightly different approximations for a finite number of rectangles but all converge to the same definite integral. They are a crucial pedagogical tool for introducing the concept of integration before formal definitions are presented.