Riemann sum

In mathematics, a Riemann sum is a sum that makes an approximation of the total area underneath a curve on a graph. The area can be known as the integral. It may also be used to define the integration operation. The sum is named after a German mathematician who was called Bernhard Riemann.

Definition

$Area = \sum_{i=1}^{n} f(y_{i})(x_{i}-x_{i-1})$

You divide the horizontal length under the part of the function you want to evaluate into "n" equal pieces. That is the n on top of the Σ (Greek letter sigma). The (x_i-x_i-1) represents the size of one horizontal segment that is created from dividing the whole by the "n". The f(y_i) is a y value in an "n" segment. Since the area of a rectangle is length × width, the multiplication of x_i and f(y_i) is the area of a rectangle for that part of the graph. The Σ means we add up all of these small rectangles to get an approximation of the area under the segment of a function.

Other websites

• A simulation showing the convergence of Riemann sums