Problem:
Find the area A bounded by the graph of
y = sin(x) and the xaxis from x = 0 to x = p.
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 A. 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 A.
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  3.14159265  1.57079633 
4  1.11072073  2.68151706  1.89611890 
8  1.58153252  2.36693068  1.97423160 
16  1.79722080  2.18991988  1.99357034 
32  1.90021859  2.09656813  1.99839336 
64  1.95051100  2.04868577  1.99959839 
128  1.97535591  2.02444329  1.99989960 
256  1.98770306  2.01224674  1.99997490 
512  1.99385780  2.00612964  1.99999372 
1024  1.99693047  2.00306640  1.99999844 

It appears that the averages of the two sums may have 2 as "its limit".