Visual Calculus - Areas

Problem:
Find the area A bounded by the graph of y = sin(x) and the x-axis from x = 0 to x = p.

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 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.

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".