Problem:
Estimate the area under the graph of a parabola y = x2 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 equal-sized 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 equal-sized 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.

nSum of areas of
inscribed rectangles
Sum of areas of
circumscribed rectangles
Average of the Two Sums
20.125000000.625000000.37500000
40.218750000.468750000.34375000
80.273437500.398437500.33593750
160.302734380.365234380.33398438
320.317871090.349121090.33349609
640.325561520.341186520.33337402
1280.329437260.337249760.33334351
2560.331382750.335289000.33333588
5120.332357410.334310530.33333397
10240.332845210.333821770.33333349

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