Home -> Overview of animations -> Fundamentals of integral calculus

Fundamentals of integral calculus

The animation shows how an integral can be approximated by establishing upper and lower sums.

Tip: The shape of the curve can be modified by moving the gray circles on either side.

To display the master function, click the content area.

The values in the upper area in the table are adjustable.

The animation was originally developed as a offline application for Windows and Macintosh systems and for integration into Microsoft PowerPoint slides. The online version presented here serve as a preview. The exe or app file can be downloaded free of charge from members area.

Download Windows

Download Macintosh


The animation shows a curve whose form can be changed by dragging the mouse. To change the shape of the curve, move the slope points of the curve on both sides.

The area below the curve is divided into several sectors (rectangles). These are the lower and upper sums. The upper sums extend beyond the curve. The lower sums are always below the curve.

Parameters for the lower and upper sums are adjustable. To accomplish this, the corresponding entries in the upper area of the table can be changed.

The number of subdivisions can also be changed. The higher the number of subdivisions, the smaller the increments in the upper and lower sums. With a very high number of subdivisions, there seems to be no longer any difference in the shape of the curve. The total area then corresponds essentially to the integral.

The second table illustrations the calculated values for the total surface area of the upper sums, the lower sums, and finally the integral itself. By increasing the number of subdivisions, the value of the total surface of all the upper sums and of the lower sums approaches the value of the integral. The upper sums approach from above, the lower sums from below to the value of the integral.

General information

Title:Fundamentals of integral calculus
Target group:
  • Teachers and lecturers
  • Independent learners
Platforms (primary):
  • Microsoft® Windows®
  • Apple® Macintosh®
  • Resizable without the loss of visual clarity
  • No installation required
DocumentsLicensing Information

Further animations for PCs

Die Animation zeigt ein Objekt, dass sich entsprechend einer Positions-Funktion bewegt. Die Funktion kann interaktiv verndert werden.
Basic principles of differential and integral calculus

Die Animationn zeigt ein rechtwickliges Dreieck. Ein Eckpunkt kann verschoben werden. Die Quadrate, die aus den drei Seiten gebildet werden, passen sich automatisch an.
Pythagorean theorem

Die Animation zeigt ein rechtwickliges Dreieck. Der Eckpunkt kann entlang eines Kreises verschoben werden. Die entsprechenden Funktionswerte werden auf der Sinus- und Kosinuskirve markiert.
Sine and cosine theorem

Further animations for smartphones/tablets

Basic principles of differential and integral calculus

Geometric relationships on a square


Authoring tool: Adobe Animate


You found a mistake and want to help us improve? Please let us know. We would also appreciate any suggestions to improve text and/or translation.

* Not mandatory