The Riemann sum is the name given to the approximate calculation of a definite integral, through a discrete summation with a finite number of terms. A common application is the approximation of the area of ​​functions on a graph.

It was the German mathematician Georg Friedrich Bernhard Riemann (1826-1866) who first offered a rigorous definition of the integral of a function in a given interval. He made it known in an article published in 1854.

Figure 1. The Riemann sum is defined on a function f and on a partition in the interval. Source: Fanny Zapata.

The Riemann sum is defined on a function y = f (x), with x belonging to the closed interval. On this interval, a partition P of n elements is made:

P = {x ₀ = a, x ₁, x ₂,…, x _n = b}

This means that the interval is divided as follows:

x _k-1 ≤ t _k ≤ x _k

Figure 1 graphically shows the Riemann sum of the function f in the interval on a partition of four subintervals, the gray rectangles.

The sum represents the total area of ​​the rectangles and the result of this sum numerically approximates the area under the curve f, between the abscissa x = x _{0 and} x = x ₄.

Of course, the approximation to the area under the curve improves greatly as the number n of partitions is greater. In this way the sum converges to the area under the curve, when the number n of partitions tends to infinity.

Formulas and properties

The Riemann sum of the function f (x) on the partition:

P = {x ₀ = a, x ₁, x ₂,…, x _n = b}

Defined over the interval, it is given by:

S (P, f) = ∑ _{k = 1} ⁿ f (t _k) (x _k - x _k-1)

Where t _k is a value in the interval. In the Riemann sum, regular intervals of width Δx = (b - a) / n are usually used, where a and b are the minimum and maximum values ​​of the abscissa, while n is the number of subdivisions.

In that case the Riemann right sum is:

Sd (f, n) = * Δx

Figure 2. Riemann right sum. Source: Wikimedia Commons. 09glasgow09.

While the Riemann left sum is expressed as:

If (f, n) = * Δx

Figure 3. Left Riemann sum. Source: Wikimedia Commons. 09glasgow09

Finally the central Riemann sum is:

Sc (f, n) = * Δx

Figure 4. Intermediate Riemann sum. Source: Wikimedia Commons. 09glasgow09

Depending on where the point t _k is located in the interval, the Riemann sum can overestimate or underestimate the exact value of the area under the curve of the function y = f (x). In other words, the rectangles can either protrude from the curve or be slightly below it.

The area under the curve

The main property of the Riemann sum and from which its importance derives, is that if the number of subdivisions tends to infinity, the result of the sum converges to the definite integral of the function:

Solved exercises

- Exercise 1

Calculate the value of the definite integral between a = -2 through b = +2 of the function:

f (x) = x ²

Make use of a Riemann sum. To do this, first find the sum for n regular partitions of the interval and then take the mathematical limit for the case that the number of partitions tends to infinity.

Solution

These are the steps to follow:

-Firstly, the partition interval is defined as:

Δx = (b - a) / n.

-Then the Riemann sum on the right corresponding to the function f (x) looks like this:

-And then it is carefully substituted in the summation:

-The next step is to separate the summations and take the constant quantities as a common factor of each sum. It is necessary to take into account that the index is i, therefore the numbers and terms with n are considered constant:

-Each sum is evaluated, since for each of them there are appropriate expressions. For example, the first of the sums gives n:

-Finally, the integral to be calculated is:

The reader can check that this is the exact result, which can be obtained by solving the indefinite integral and evaluating the limits of integration by Barrow's rule.

- Exercise 2

Approximately determine the area under the function:

f (x) = (1 / √ (2π)) e ^{(-x 2 /2)}

Enter x = -1 and x = + 1, using a central Riemann sum with 10 partitions. Compare with the exact result and estimate the percentage difference.

Solution

The step or increment between two successive discrete values ​​is:

Δx = (1 - (-1) / 10 = 0.2

So the partition P on which the rectangles are defined looks like this:

P = {-1.0; -0.8; -0.6; -0.4; -0.2; 0.0; 0.2; 0.4; 0.6; 0.8; 1.0}

But since what is wanted is the central sum, the function f (x) will be evaluated at the midpoints of the subintervals, that is, in the set:

T = {-0.9; -0.7; -0.5; -0.3; -0.1; 0.1; 0.3; 0.5; 0.7; 0.9}.

The (central) Riemann sum looks like this:

S = f (-0.9) * 0.2 + f (-0.7) * 0.2 + f (-0.5) * 0.2 +… + f (0.7) * 0.2 + f (0.9) * 0.2

Since the function f is symmetric, it is possible to reduce the sum to only 5 terms and the result is multiplied by two:

S = 2 * 0.2 * {f (0.1) + f (0.3) + f (0.5) + f (0.7) + f (0.9)}

S = 2 * 0.2 * {0.397+ 0.381+ 0.352+ 0.312+ 0.266} = 0.683

The function given in this example is none other than the well-known Gaussian bell (normalized, with mean equal to zero and standard deviation one). The area under the curve in the interval for this function is known to be 0.6827.

Figure 5. Area under a Gaussian bell approximated by a Riemann sum. Source: F. Zapata.

This means that the approximate solution with just 10 terms matches the exact solution to three decimal places. The percentage error between the approximate and the exact integral is 0.07%.

References

1. Casteleiro, JM, & Gómez-Álvarez, RP (2002). Integral calculus (Illustrated ed.). Madrid: ESIC Editorial.
2. Unican. History of the concept of integral. Recovered from: repositorio.unican.es
3. UIS. Riemann sums. Recovered from: matematicas.uis.edu.co
4. Wikipedia. Riemann sum. Recovered from: es.wikipedia.com
5. Wikipedia. Riemann integration. Recovered from: es.wikipedia.com