The yellow region shown in the figure below has four sides.
We are interested in finding the area of this region. We can make an estimate based on the picture below.
In this picture we divide the yellow region into four strips, each of which has width 0.25. Using the notation
x(i) = 1 + 0.25 i
we see that the i-th strip goes from x(i - 1) to x(i). For example, the second strip goes from 1.25 to 1.50.
We estimate the area of each strip by looking at a rectangle whose height is determined by the right edge of the strip. For example, the area of the second strip is estimated by looking at a rectangle whose height is
(1.50)^2 - 2 (1.50) + 2 = 1.25
Thus, the area of this strip is approximately
Area = (height) (width) = (1.25) (.25) = 0.3125
The same procedure is used to estimate the area of each strip. Then the total area is estimated by adding up the estimates for the four strips. The necessary calculations are shown in your CAS window.
Notice that this region is one-fourth of a circle of radius 2 and, thus, its area is pi.
Now we return to our example above. Notice that for this particular example each rectangle overestimates the area of its strip. The overestimates are indicated by the red regions in the graph below.
Notice that each of the four overestimate pieces has the same width as the strips -- namely 0.25. Notice also that the overestimates look like stairs -- they can be placed on top of each other because the bottom of each piece is the top of the piece to its left.
