Riemann sum calculator with steps and graph

Riemann sum calculator

L5 = Δx ( �(x0) + �(x1) + �(x2) + �(x3) + �(x4)
f(x) =
Interval: x
Number of Subintervals n =
Δx = (b � a) � n
= (3.1416 � 0) � 5
≈ 0.62832


The online Riemann Sum calculator is an excellent resource for all those students who are studying the subject of Calculus. With this calculator you will be able to solve Riemann Sums of all kinds of functions of a single variable. To do this, it uses 7 different methods:

  1. Left Riemann sum
  2. Midpoint Riemann sum
  3. Right Riemann sum
  4. Random point
  5. Trapezoidal rule
  6. Simpson’s method
  7. Adaptive Simpson’s method

Instructions for using the Riemann Sums calculator

To use this calculator you must follow these simple steps:

  1. Enter the function in the field that has the label f(x)= to its left. To enter the function you must use the variable x, it must also be written using lowercase.
  2. Enter the interval for which you will perform the Riemann sum calculation.
  3. Enter the value of n, which indicates the number of subintervals that will be used.

The graph will automatically be generated and the numerical result of the sum for each of the 7 methods mentioned above will be displayed. You will be able to see the resulting graph and the development of each method just by selecting its corresponding checkbox.

What is the Riemann sum?

In mathematics, the Riemann sum is a numerical integration method that helps us calculate the approximate value of a definite integral, that is, the area under a curve for a given interval. This method is very useful when it is not possible to use the Fundamental Theorem of Calculus.

riemann sum

Let f(x) be a continuous function defined on the closed interval [a,b], S is the region under the curve y=f(x) on the indicated interval.

Suppose we partition the interval [a,b] into n equally spaced subintervals. Each subinterval has an amplitude equal to Δx=(b−a)/n.

riemann sum example 01

Choosing a representative point of each interval x1,x2,…,xn, we can roughly calculate the area under the curve using the following formula:
​ .

riemann sum formula

The above expression is called the Riemann Sum. So we can define the area under the curve as the limit of the Riemann Sum with the number of subintervals n tending to infinity.

riemann sum formula 02

The above limit is known as the Riemann Integral, since it is widely used as a numerical method to approximate definite integrals.

Methods for solving Riemann Sums

There are different methods to calculate the Riemann sums, below we will present the most used:

Left Riemann sum

Left Riemann sum

In this method, the left end of the rectangles of each subinterval are those that touch the curve, as can be seen in the previous image. The base of each rectangle will be equal to Δx and the height will be given by f(a+iΔx).

The Riemann sum formula would be:

formula for left riemann sum

Right Riemann sum

right Riemann sum

Unlike the previous method, here it is the right end of the rectangles of each subinterval that touches the curve, as can be seen in the image above. The base and height of each rectangle are the same as in the previous method, Δx and f(a+iΔx) respectively. The formula for the Riemann sum on the right hand side is:

Midpoint Riemann sum

midpoint riemann sum

In this case, the midpoint of the rectangle of each subinterval will be the one that will touch the curve of the function. The base and height of each rectangle are the same as in the previous methods, Δx and f(a+iΔx) respectively.

The summation formula following the midpoint rule is:

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