Riemann sum illustration

The GeoGebra worksheet below illustrates the use of Riemann sums to approximate the area under a curve. In this case, the curve is the top half of a circle of radius 2. The area to be computed is the area of a quarter circle. We know from grade school that the area of the whole circle should be πr², where r=2, and so the area of the whole circle should be 4π. Thus, the area of the quater circle should be one fourth of this, or just π. By using more rectangles, you can get a better and better idea what the value of π is. However, even with 2000 rectangles, the upper and lower sums don't agree perfectly well, because of how steep the graph is near x = 2.

Here are the commands used to create this worksheet:

f(x) = sqrt(4-x^2)
n = 12
Right click n under Free Objects, Show Object.
Right click n under Free Objects, Properties, Slider, min 1, max 2000, increment 1, width 400.
L = LowerSum[f,0,2,n]
U = Uppersum[f,0,2,n]
Options, Rounding, 15 decimal places
Right click L, de-select Show Label
Right click U, de-select Show Label

Craig L. Zirbel, Created with GeoGebra