Riemann Sum

A Riemann sum is a mathematical technique used to approximate the definite integral of a function over a specified interval. Named after the German mathematician Bernhard Riemann, Riemann sums provide a way to estimate the area under a curve using a series of rectangles or other geometric shapes.

To compute a Riemann sum, the interval over which the integral is to be computed is divided into smaller subintervals, typically of equal width. The function’s value is then sampled at a specific point within each subinterval, which determines the height of the rectangle. The sum of the areas of all these rectangles provides an approximation of the integral.

There are several types of Riemann sums, depending on the choice of the sample point within each subinterval. The most common types are the left Riemann sum, the right Riemann sum, and the midpoint Riemann sum. In each case, the sample points are chosen as the left endpoint, the right endpoint, or the midpoint of each subinterval, respectively.

As the number of subintervals increases and their width decreases, the Riemann sum approaches the exact value of the integral. The convergence of Riemann sums to the true integral value is formalized by the Fundamental Theorem of Calculus. Riemann sums are an essential tool for understanding the concept of integration and serve as the basis for more advanced numerical integration techniques such as the Trapezoidal Rule and Simpson’s Rule. These methods provide more accurate approximations of definite integrals by using more sophisticated geometric shapes or higher-order polynomials to approximate the area under the curve.