mittenpunktsregeln

The mittenpunktsregeln, often translated as the "midpoint rule," is a numerical integration technique used to approximate the definite integral of a function. It is a simple yet effective method, particularly useful for functions where analytical integration is difficult or impossible. The rule is based on approximating the area under the curve of a function over a given interval by using rectangles whose heights are determined by the function's value at the midpoint of each subinterval.

To apply the mittenpunktsregeln, the interval of integration is first divided into a number of equal subintervals. For each subinterval, the midpoint is calculated. The function is then evaluated at this midpoint. The area of the rectangle for that subinterval is found by multiplying the function's value at the midpoint by the width of the subinterval. Summing the areas of all these rectangles provides an approximation of the total definite integral.

The accuracy of the mittenpunktsregeln generally increases with the number of subintervals used. It is considered a specific case of Newton-Cotes formulas, which are a family of methods for numerical integration. Compared to other basic methods like the trapezoidal rule, the midpoint rule often provides a more accurate approximation for the same number of subintervals due to its centered evaluation point. This makes it a valuable tool in various fields requiring numerical solutions to integration problems, such as physics, engineering, and economics.