Where a and b are the endpoints and n is the number of partitions (rectangles in this case). Since equally spaced rectangles are used in this example, Δx is the same for each partition, and we can use the following expression to determine Δx: The Riemann sum, S, can be expressed as follows: The figure below depicts a left Riemann sum for f(x) = x 2 over the interval the region is partitioned using 6 rectangles of equal width. The left Riemann sum involves approximating a function through use of its left endpoint this means that the left endpoint of the partition is the point that intersects the curve. One of the ways in which Riemann sums differ is the choice of, which affects the point at which the curve intersects the partition. There are a number of different types of Riemann sums. Note that since Δx approaches 0, it does not matter whether a left, right, or midpoint Riemann sum is used the limit will still have the same value regardless which point is chosen since they all converge. In other words, the limit of a Riemann sum as Δx approaches 0 over an interval is equal to the definite integral over that interval, or: Rather, the sum would be the exact value of the area under the curve. Assuming that it were possible to achieve an infinite number of partitions, the Riemann sum would no longer be an approximation. As the number of partitions approaches infinity, Δx approaches 0. Riemann sums are used to approximate definite integrals the larger the number of partitions (n), the more accurate the approximation. for each type of Riemann sum is depicted in the figure below: Thus, represents the area of a given rectangle in the Riemann sum, and the choice of determines which type of Riemann sum (left, right, or midpoint) is being used. Where represents the width of the rectangles ( ), and is a value within the interval such that is the height of the rectangle. Definition of a Riemann sumĪ Riemann sum is defined using summation notation as follows This is the basis of the definition of a Riemann sum. For example, the area under the curve from x = 0 to x = 3 is estimated in each of the graphs below using 4, 10, and 20 rectangles.Īs can be seen from the figure, the higher the number of rectangles used, the more accurate the representation of the area under the curve. This error can be reduced by using a larger number of shapes with smaller width the smaller the width of the rectangles, the more closely they can represent the shape of the region, and the more accurate the estimation of the area. Since the shapes being used won't exactly match the shape of the region, there will be some error. The area of these shapes are then added to estimate the area under the curve most typically, rectangles of even spacing and width are used, though this does not have to be the case. It is used to estimate the area under a curve by partitioning the region into shapes (whose areas are typically simple to compute) similar to the region being measured. In calculus, the Riemann sum is commonly taught as an introduction to definite integrals. Home / calculus / integral / riemann sum Riemann sumĪ Riemann sum is a method used for approximating an integral using a finite sum.
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