Problem:
Estimate the area under the graph of a parabola y = x^{2} from
x = 0 to x = 1.
Visualization:
[Press here to see animation again!]
In the animation above, first you can see how by increasing the number of
equalsized intervals the sum of the areas of inscribed rectangles can better
approximate the area. Then this is followed by showing how by
increasing the number of equalsized intervals the sum of the areas of
circumscribed rectangles can better approximate the area.
The following table indicates the sums of the various areas together with their
averages. n indicates the number of rectangles.
n  Sum of areas of inscribed rectangles 
Sum of areas of circumscribed rectangles 
Average of the Two Sums 
2  0.12500000  0.62500000  0.37500000 
4  0.21875000  0.46875000  0.34375000 
8  0.27343750  0.39843750  0.33593750 
16  0.30273438  0.36523438  0.33398438 
32  0.31787109  0.34912109  0.33349609 
64  0.32556152  0.34118652  0.33337402 
128  0.32943726  0.33724976  0.33334351 
256  0.33138275  0.33528900  0.33333588 
512  0.33235741  0.33431053  0.33333397 
1024  0.33284521  0.33382177  0.33333349 

It appears that the averages of the two sums may have 1/3 as "its limit".