Visual Calculus - Areas

Problem:
Estimate the area under the graph of a parabola y = x² from x = 0 to x = 1.

Visualization:

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.

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".