Riemann Sums

In mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It may also be used to define the integration operation. The method was named after German mathematician Bernhard Riemann.

For a function f: D --> R, where D is a subset of the real numbers R, I = [a, b] is 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 P = {[x₀, x₁), [x₁, x₂), ... [x_n-1, x_n]} of I. Because P is a partition with n elements of I, 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. The average of the left and right Riemann sum is the trapezoidal sum. If it is given that

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.

Any Riemann sum on a given partition (that is, for any choice of y_i between x_i 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.

Methods

The four methods of Riemann summation are usually best approached with partitions of equal size. The interval [a, b] is therefore divided 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 sum

For the left Riemann sum, approximating the function by its value at the left-end point 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

         Q [ f(a) + f(a+Q) + f(a+2Q) + . . .+ f(b-Q) ]

The left Riemann sum amounts to an overestimation if f is monotonically decreasing on this interval, and an underestimation if it is monotonically increasing.

Right sum

f is here approximated 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 produces

           Q [ f(a+Q) + f(a+2Q) + . . .+ f(b) ]

The right Riemann sum amounts to an overestimation if f is monotonically increasing, and an underestimation if it is monotonically decreasing.

Middle sum

Approximating f at the midpoint of each interval gives f(a + Q/2) for the first interval, for the next one f(a + 3Q/2), and so on until f(b-Q/2). Summing up the areas gives

The error of this formula will be

where M2 is the maximum value of the absolute value of on the interval.

Trapezoidal rule

In this case, the values of the function f on an interval are 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 produces

where M2 is the maximum value of the absolute value of