In mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space It may also be used to define the integration operation. The sums are named after the German mathematician Bernhard Riemann.

Contents 1 Definition 2 Methods 2.1 Left Riemann sum 2.2 Right Riemann sum 2.3 Middle sum 2.4 Trapezoidal rule 3 See also 4 External links

Definition

Consider a function f: D → R, where D is a subset of the real numbers R, and let I = [a, b] be a closed interval contained in D. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, the real numbers may be described informally in several different ways In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set A finite set of points {x₀, x₁, x₂, . . . x_n} such that a = x₀ < x₁ < x₂ . . . < x_n = b creates a partition

P = {[x₀, x₁), [x₁, x₂), . In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks " . . [x_n-1, x_n]}

of I.

If P is a partition with n elements of I, then the Riemann sum of f over I with the partition P is defined as

where x_i-1 ≤ y_i ≤ x_i. The choice of y_i in this interval is arbitrary. If y_i = x_i-1 for all i, then S is called a left Riemann sum. If y_i = x_i, then S is called a right Riemann sum. If y_i = (x_i+x_i-1)/2, then S is called a middle Riemann sum. By averaging the left and right Riemann sum one obtains the so-called trapezoidal sum.

Suppose we have

where v_i is the supremum of f over [x_i-1, x_i]; then S is defined to be an upper Riemann sum. Similarly, if v_i is the infimum of f over [x_i−1, x_i], then S is a lower Riemann sum. In Mathematics the infimum of a Subset of some set is the Greatest element, not necessarily in the subset that is less than or equal to all elements of

Any Riemann sum on a given partition (that is, for any choice of y_i between x_i-1 and x_i) is contained between the lower and the upper Riemann sums. A function is defined to be Riemann integrable if the lower and upper Riemann sums get ever closer as the partition gets finer and finer. In the branch of Mathematics known as Real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the Integral This fact can also be used for numerical integration. In Numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite Integral, and by extension

Methods

As stated above, there are four common methods to compute a Riemann sum: left, right, middle, and trapezoidal. We will elaborate on them in the simple case when the partition is made up of intervals of equal size. Thus, divide the interval [a, b] into n subintervals, each of length Q = (b − a) / n. The points in the partition will then be

a, a + Q, a + 2Q, . . . , a + (n−2)Q, a + (n−1)Q, b.

Left Riemann sum

For the left Riemann sum, we will approximate the function by its value at the left-end point. This gives multiple rectangles with base Q and height f(a + iQ). Doing this for i = 0, 1, . . . , n−1, and adding up the resulting areas gives us

The left-hand Riemann sum will be an overestimation if f is monotonically decreasing on this interval, and an underestimation if it is monotonically increasing.

Right Riemann sum

Here, for each interval we will approximate f by the value at the right endpoint. This gives multiple rectangles with base Q and height f(a + iQ). Doing this for i = 1, 2, . . . , n−1, n, and adding up the resulting areas gives us

The right-hand Riemann sum will be an overestimation if the function f is monotonically increasing, and an underestimation if it is monotonically decreasing.

Middle sum

In this case we will take as approximation for f in each interval its value at the midpoint. For the first interval we will thus have f(a + Q/2), for the next one f(a + 3Q/2), and so on until f(b-Q/2) is reached. Summing up the areas, we find

The error of this formula will be

where M₂ is the maximum value of the absolute value of on the interval. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.

Trapezoidal rule

In this case, the values of the function f on an interval will be approximated by the average of the values at the left and right endpoints. In the same manner as above, a simple calculation using the area formula A = h(b₁ + b₂) / 2 for a trapezium with parallel sides b₁, b₂ and height h one calculates the Riemann sum to be

The error of this approximation for the integral is

where M₂ is the maximum value of the absolute value of

See also

• Riemann-Stieltjes integral
• Lebesgue integral
• Simpson's rule

External links

• A simulation showing the convergence of Riemann sums

A trapezoid (in North America or a trapezium (in Britain and elsewhere is a Quadrilateral (a closed plane shape with four linear sides that has at least one In Mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes In Mathematics, the Integral of a non-negative function can be regarded in the simplest case as the Area between the graph of In Numerical analysis, Simpson's rule is a method for Numerical integration, the numerical approximation of Definite integrals Specifically it is the following

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