Content: This page contains an applet illustrating the concept of Riemann sums for functions in a single variable and instructions for its use.


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Analysis for Computer Scientists, Chapter 11


Entering the integrand

To visualise the approximate integration of a function by means of Riemann sums, a function must be defined in the field f(x)= (detailed information on the syntax can be found here). The interval of integration is set in the field Interval in the format [a,b].


Riemann sums


The number of subdivisions in the interval is specified in the field Subdivisions. To visualise the Riemann sum press the button Compute. In the standard configuration, the intermediate points for the function evaluations are selected at random.

Configuring the subdivisions

After drawing the function graph and the approximating rectangles, the subdivision and the intermediate points can be changed with the mouse in the following way:

  • Drag a point while pressing the left mouse button: Move a point;
  • Click at a point with the third mouse button: Delete the point;
  • Click at a free place on the x-axis with the right mouse button: Add a subdivision point.


In the output field (to the right of the function graph) the number of subdivisions, the value of the Riemann sum, the exact value of the integral (computed by a 5-point Gaussian quadrature rule) and the error of the Riemann sum are given.


Riemann output


The option Activate intervals in the tab Options determines if the subdivision can be changed with the mouse. In addition, the option Neg./pos.summands in the same colour specifies if the rectangles below and above the x-axis are highlighted in the same colour. Finally, the option Random subdivision determines if the subdivision of the integration interval is chosen randomly or if an equidistant subdivision is used.


If you have further questions or comments, or if you found a bug, please send us an e-mail.

 Financially supported by

University of Innsbruck: New Media and Learning Technologies
Austrian Federal Ministry of Education, Science and Research

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