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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.
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.
Title:  Fundamentals of integral calculus 

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Basic principles of differential and integral calculus
Basic principles of differential and integral calculus
Geometric relationships on a square
Authoring tool: Adobe Animate
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