We divide [0,4] into four sub-intervals of width 1. The first trapezoid looks like this, with heights *f* (0) = 1 and *f* (1) = 2. It has area The second trapezoid looks like this, with heights 2 (we know this from the previous trapezoid) and *f** *(2) = 5. It has area The third trapezoid has heights 5 (known from previous trapezoid) and *f* (3) = 10. It has area . The third trapezoid has heights 10 (known from previous trapezoid) and *f* (4) = 17. It has area Adding up the areas of the trapezoids, we estimate the area of *R* to be 1.5 + 3.5 + 7.5 + 13.5 = 26. This is an overestimate, since again the trapezoids cover all of *R* and then some. |