Problem:
Find the area A bounded by the graph of y = sin(x) and the x-axis 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 equal-sized 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 equal-sized 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.

nSum of areas of
inscribed rectangles
Sum of areas of
circumscribed rectangles
Average of the Two Sums
203.141592651.57079633
41.110720732.681517061.89611890
81.581532522.366930681.97423160
161.797220802.189919881.99357034
321.900218592.096568131.99839336
641.950511002.048685771.99959839
1281.975355912.024443291.99989960
2561.987703062.012246741.99997490
5121.993857802.006129641.99999372
10241.996930472.003066401.99999844

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