Riemannszummák

Riemannszummák, often translated as Riemann sums, are a fundamental concept in calculus used to define and calculate definite integrals. They provide a way to approximate the area under a curve by dividing the region into a series of thin rectangles. The width of each rectangle is determined by partitioning the interval over which the integral is being calculated, and the height is determined by the function's value at a specific point within that subinterval.

There are several types of Riemann sums, distinguished by the point chosen to determine the rectangle's height. Common examples include left Riemann sums, where the height is the function's value at the left endpoint of the subinterval; right Riemann sums, where the height is taken at the right endpoint; and midpoint Riemann sums, where the height is determined by the function's value at the midpoint of the subinterval. Another variation is the upper or lower Riemann sum, which uses the maximum or minimum value of the function within the subinterval, respectively.

The accuracy of a Riemann sum approximation increases as the number of rectangles, or partitions, grows larger. As the width of each rectangle approaches zero, the sum of the areas of these rectangles converges to the exact value of the definite integral. This limit process is the basis for the formal definition of the definite integral. Riemann sums are essential for understanding the relationship between discrete summation and continuous integration.