Riemann sum calculator with steps and graph - Theory and examples

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

1. Input the Function:

   Enter the mathematical function in the input box labeled f(x). Example: x^2 + 4.

2. Define the Interval:

   Set the lower and upper bounds of the interval in the respective input fields. Example: 0 to pi.

3. Set the Number of Subintervals:

   Specify the number of subintervals n. This determines the precision of the approximation. Example: 5.

4. Select the Calculation Methods:

   Choose one or more methods by checking the corresponding boxes:

   • Left Riemann Sum: Uses left endpoints.
   • Midpoint Riemann Sum: Uses midpoints.
   • Right Riemann Sum: Uses right endpoints.
   • Random Point: Randomly selects points.
   • Trapezoidal Rule: Uses trapezoidal approximations.
   • Simpson's Method: Uses quadratic polynomials for higher accuracy.
   • Adaptive Simpson's Method: Dynamically adjusts intervals for higher precision.
5. Set Decimal Precision:

   Define the number of decimal places for results in the Decimals input field. Example: 4.

6. Generate Results:

   Click the "Calculate" button. The graph will automatically be generated, and the numerical results for each selected method will be displayed. You can view the graph and step-by-step development of each method by selecting its checkbox.

7. View Supported Functions:

   Click the "?" button to display a modal containing the list of supported mathematical functions.

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.

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.

Choosing a representative point of each interval x₁,x₂,...,x_n, we can roughly calculate the area under the curve using the following 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.

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

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:

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

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: