Find the area A bounded by the graph of
y = sin(x) and the x-axis from x = 0 to x = p.
[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.
|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 2 as "its limit".