Estimate the area under the graph of a parabola y = x2 from
x = 0 to x = 1.
[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.
|n||Sum of areas of
|Sum of areas of
|Average of the Two Sums|
It appears that the averages of the two sums may have 1/3 as "its limit".