-The concept of upper sum and lower sum is commonly applied in calculating the area under a curve or area represented by a function or graph.[br]-we make approximation of the area of function by using rectangles.[br]-Upper sum [math]\Longrightarrow[/math] The height of rectangles is absolute maximum of f(x) on the sub interval [A,B].[br]-Lower sum[math]\Longrightarrow[/math] The height of rectangles is absolute minimum of f(x) on the sub interval [A,B].

Dynamic applet of Lower and Upper sum of function f(x) with interval [A,B].

- you can see in this applet, a function with its lower sum and upper sum on the in the interval A,B].[br]- Here, upper sum denote the maximum area of the function f(x) between the interval [A,B] and lower sum denote the minimum area of the function f(x) between the interval [A,B].[br]- Now, move slider then you can see the number of rectangles n changes and upper sum,lower sum and difference of upper and lower sum also changes its values.[br]-So, if the number of rectangles n increases then upper sum is decrease ,lower sum increase and difference of upper and lower sum also decrease.[br] And if the number of rectangles n decrease then upper sum is increase ,lower sum decrease and difference of upper and lower sum is increases.[br]finally; conclude that, the upper sum and lower sum is depend on the number of rectangles we take and the interval we decide.

Construction steps of this applet are as follows:-[br]1.Enter a cubic polynomial f(x)=-0.5x3+2x2-x+1 by using input bar.[br]2.create two points A and B on the X-axis (which is interval of the function f(x)) by using point tool bar.[br]3.Create slider for the number n with interval 1 to 50 with increment 1.[br]4. Enter the Upper sum= upper sum[f,X(A),X(B),n] by using input bar.[br] here, X(A) gives the coordinate of point A. Number n determines the number of rectangles used in order to calculate the lower sum and upper sum.[br]5.Enter lower sum= Lower sum[f,X(a),X(B),n].[br]6.Insert the dynamic text Upper sum = and select upper sum from the objects.[br]7.Insert the dynamic text Lower sum = and select lower sum from the objects.[br]8. calculate the Difference = Upper sum -Lower sum by using input bar.[br]9. Insert the dynamic text Difference = and select Difference from the objects.[br]10.Enter F- integral [f,X(A),X(B) by using input bar.[br]11.Insert dynamic text Integral- and select F from the objects .[br] At last, fix slider and decorate your applet by using object properties it helps to make applet more attractive and save your ggb file.