In mathematics, a Riemann sum is a method for approximating the values of integrals. It may also be used to define the integration operation. The sums are named after Bernhard Riemann. For other meanings of mathematics or math, see mathematics (disambiguation). ... In calculus, the integral of a function is a generalization of area, mass, volume and total. ... Summation is the addition of a set of numbers; the result is their sum. ... Bernhard Riemann. ...

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. A finite set of points {x₀, x₁, x₂, ... x_n} such that a = x₀ < x₁ < x₂ ... < x_n = b creates a partition Partial plot of a function f. ... In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ... A partition of U into 6 blocks: a Venn diagram representation. ...

P = {[x₀, x₁), [x₁, x₂), ... [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 supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ... In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ...

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. This fact can also be used for numerical integration. If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ... Numerical Integration with the Monte Carlo method: Nodes are random equally distributed. ...

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

A left Riemann sum of x³ over [0,2] using 4 subdivisions.

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 Image File history File links LeftRiemann2. ... Image File history File links LeftRiemann2. ...

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. In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ...

Right Riemann sum

A Right Riemann sum of x³ over [0,2] using 4 subdivisions.

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 Image File history File links RightRiemann2. ... Image File history File links RightRiemann2. ...

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

A middle Riemann sum of x³ over [0,2] using 4 subdivisions.

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 Image File history File links MidRiemann2. ... Image File history File links MidRiemann2. ...

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 modulus1) of a real number is its numerical value without regard to its sign. ...

Trapezoidal rule

A trapezoidal Riemann sum of x³ over [0,2] using 4 subdivisions.

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 Image File history File links TrapRiemann2. ... Image File history File links TrapRiemann2. ... In acrobatics, the trapeze is a certain acrobatic device that is shaped like a trapezoid. ...

The error of this approximation for the integral is

where M₂ is the maximum value of the absolute value of

See also

• Riemann integral
• Riemann-Stieltjes integral

If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ... In mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. ...

External links

• A simulation showing the convergence of Riemann sums