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The animation shows how an integral can be determined approximately by the formation of upper and lower sums.
Tip: You can change the shape of the curve interactively. Move the circles.
To display the master function, click the content
The values in the first table can be changed.
The animation was developed as a offline application for Windows and Macintosh systems and for integration into Microsoft PowerPoint slides. The online version presented here serves as a preview. The exe or app file can be downloaded free of charge in the member area.
The downloaded file can be included in PowerPoint.
The animation shows a curve whose form can be interactively changed with the mouse. To change the shape of the curve, the slope points of the curve must be shifted at the beginning and at the end.
The area below the curve is divided into several rectangular segments. These are the lower and upper sums, ie the rectangles whose heights are defined by the curve. The sum totals extend beyond the curve. The subsumptions are always below the curve.
The limits of the lower and upper sums are adjustable. For this purpose, the corresponding entries must be changed in the upper table.
The number of subdivisions can also be changed. The higher the number of subdivisions, the finer the gradations of the upper and lower sums. With a very high number of subdivisions, there is no longer any difference to the shape of the curve. The total area then corresponds almost to the integral, ie to the exactly calculated area below the curve.
The second table shows the calculated values for the total surface area of the sumums, the subtotals, and finally the integral itself. By increasing the number of subdivisions, it can be observed that the value of the total surface of all the sumums and also of the subtotals approaches the value of the integral , The sumums approach from above, the subsumptions from below to the value of the integral.
Title:  Basis of the integral calculus 

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About the security of the Flash Player 
Basic principles of differential and integral calculus
Basic principles of differential and integral calculus
Geometric relationships at the quadrangle
Authoring tool: Adobe Animate
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