Calculation of an integral defined by the Riemann sums

In this lesson I’m going to explain how to get the expression to calculate a defined integral using Riemann sums.

Geometric interpretation of Riemann sums

An integral defined in an interval [a,b] gives us the value of the area enclosed between a function f(x) and the x-axis in an interval [a,b], as long as the function is continuous.

Another way to calculate the area enclosed below a curve would be to divide the area into equal rectangles and add the area of each of the rectangles, although this calculation would be approximate:

sumas de Riemann

If we take one of those rectangles:

sumas de riemann integral definida

The base would be the difference of two values of x and the height would be the value of the function for X=Xi

suma de riemann integral definida

The area of each rectangle would be obtained by multiplying the base by the height and it would remain:

sumas de riemann integral

If we make the rectangles smaller and smaller, the calculation of the area becomes more and more exact.

suma de riemann integral

And if rectangles are made infinitely small and we have infinite rectangles, the infinite sum of those rectangles would be the exact area of the enclosed area below that function and would be equal to the defined integral of that function for an interval [a,b]:

integrals defined sum of riemann

From where we obtain the expression used to solve integrals defined by Riemann sums, in which, as we have seen before, the area of each rectangle would be equal to:

sumas de riemann calculo integral

Where the increment value of x for an interval [a,b] will be defined as:

the sum of riemann the defined integral

And the value of Xi as:

integral defined sum of riemann exercises solved

All this is understood much more clearly with an example, which is what we will see below

Example of how to obtain the expression to calculate an integral by the sums of Riemann

Obtain the expression of the Riemann sums of the following integral:

sum of riemann integral calculation

The function in this case is:

integrals defined step by step

For an interval [0,3], therefore a=0 and b=3.

We define the increment of x for that interval:

integral definida riemann

Y with this incremental expression of x, we calculate Xi:

approximate calculation of the defined integralNow we get the function for X=Xi, substituting x for Xi in the original function:

integral de riemann ejemplo

We calculate the area of each rectangle, multiplying the expressions obtained from f(Xi) and the increment of x:

calculation of the defined integral

And we have left:

integral definida resolver

So the value of the integral by Riemann sums is:

calcululo integral sumatorias

And to obtain the result of the area, we would have to solve the limit of the sum, which we will see in the next section.

How to solve an integral for the sums of Riemann

We are going to solve the expression that was left in the previous section and therefore we will solve the integral by Riemann’s sums.

We start from the previous expression:

integral de riemann formula

We solve the parenthesis by elevating the cube:

integral defined examples step by step

And we multiply both fractions:

sumatorias calculo integral

What is constant we can take out of the summation. Anything that does not carry i is considered constant, so we take the terms that do not carry i out of the summation:

riemann calcululo integral

At this point the sum of i elevated to the curo from i=0 to n is equal to this formula:

riemann sum formulas

It can be demonstrated but it is not the objective of this lesson.

We substitute the summation by its expression, according to the previous formula:

formulas de riemann

We multiply the parenthesis:

integral definida de riemann

And we simplify terms:

sumas de riemann formula

And when we solve the limit we are left:

suma de riemann formula

The result is in square units because we are calculating an area.

To show that the result is correct, I am going to solve the integral defined by Barrow’s rule:

sumas de riemann formulas

suma de riemann e integral definida

And as it could not be otherwise, the result is the same.

Summary formula to solve Riemann sums

Finally, I leave you here the formulas of the sums from the sum of 1 to the sum of i al cubo (that we have used in the example), from 1=0 to n, that you are going to need to solve integrals with Riemann’s sums:

riemann sum formula

sumatoria de rieman

formulas de sumas de riemann

formula suma de riemann