Riemannsummor

Riemann summation is a mathematical concept that arises in the study of definite integrals. It is a method for approximating the value of a definite integral by summing the areas of small rectangles that approximate the region under the curve. The Riemann sum is named after Bernhard Riemann, who first introduced the concept in the mid-19th century.

The basic idea behind Riemann summation is to divide the region under the curve into small subintervals, or strips, and to approximate the area of each strip by the area of a small rectangle. The Riemann sum is then found by summing the areas of these small rectangles. The width of the rectangles is determined by the choice of subdivision, and the more subdivisions that are used, the more accurate the approximation will be.

The Riemann sum can be written in the form:

∑[f(x_i) Δx]

where f(x_i) is the value of the function at a point x_i in the ith subdivision, and Δx is the width of each subdivision. The sum is taken over all subintervals.

One of the key properties of the Riemann sum is that it can be made to approximate the definite integral of a function as closely as desired, as long as the number of subdivisions is increased without bound. This makes the Riemann sum a powerful tool in the study of calculus, particularly in the development of the fundamental theorem of calculus.