This worksheet examines the constructions and accuracies of different integral approximation methods and its relations with the exact integration provided by the primitive function.[br][br]The check-boxes on the left side allow you to view the exact area and to toggle between rectangles with right, left and middle points and with the trapezoidal approximations.[br]The extremes of integration [i]a[/i] and [i]b[/i] and the number of intervals [i]n[/i] can be set with the sliders in the upper left side. [br][br]The actual area and that of each approximation are shown at the bottom left side of the worksheet. The actual area can be exactly calculated with the primitive [math]Y=F\left(x\right)[/math] of the function [math]f\left(x\right)[/math] where [math]F'\left(x\right)=f\left(x\right)[/math].[br][br]It can be shown that, when [math]\Delta x\rightarrow0[/math], the area equals the definite integral[br][math]\int_a^bf\left(x\right)dx=F\left(b\right)-F\left(a\right)[/math] [br]since it is[br][math]\sum f\left(x\right)\Delta x=\sum F'\left(x\right)\Delta x=\Sigma\Delta Y=F\left(b\right)-F\left(a\right)[/math][br][br]The "Show primitive function" check-box allows to see this construction.[br]There are two alternative functions [i]f[/i][sub]1[/sub]([i]x[/i]) and [i]f[/i][sub]2[/sub]([i]x[/i]) already set in the workbook.[br]With the check-box "[i]Alternative function f[sub]2[/sub](x)[/i]" it's possible to switch from the first one to the second one.

[b]Credits[/b]: [i]Christopher Stover[/i]. This worksheet builds up from his original worksheet "Rectangular and Trapezoidal Integral Approximations", once available on [url=http://personal.bgsu.edu/~stoverc/Geogebra/index.html]http://personal.bgsu.edu/~stoverc/Geogebra/index.html[/url]