Riemannösszeg

A Riemannösszeg, or Riemann sum, is a method for approximating the area under a curve. It's a fundamental concept in calculus, particularly in the definition of the definite integral. The process involves dividing the region under the curve into a series of narrow rectangles or trapezoids. The width of each rectangle is determined by a partition of the interval over which the area is being calculated, and the height is determined by the function's value at a specific point within that subinterval.

There are several common types of Riemann sums. The left Riemann sum uses the function's value at the left endpoint of each subinterval to determine the height of the rectangle. The right Riemann sum uses the right endpoint. The midpoint Riemann sum uses the midpoint of each subinterval. The trapezoidal rule approximates the area using trapezoids instead of rectangles, which can often provide a more accurate approximation for the same number of subdivisions.

As the number of rectangles or trapezoids increases, and their width consequently decreases, the Riemann sum converges to the exact area under the curve. This limit is precisely what defines the definite integral of the function over the given interval. The Riemann sum thus provides a constructive way to understand and calculate the area, forming the basis for numerical integration techniques.