Figure (a) shows a vacant lot with a 80 ft frontage in a development. To estimate its area, we introduce a coordinate system so that the x-axis coincides with the edge of the straight road forming the lower boundary of the property, as shown in Figure (b). Then, thinking of the upper boundary of the property as the graph of a continuous function f over the interval [0, 80], we see that the problem is mathematically equivalent to that of finding the area under the graph of f on [0, 80]. To estimate the area of the lot using a Riemann sum, we divide the interval [0, 80] into four equal subintervals of length 20 ft. Then, using surveyor's equipment, we measure the distance from the midpoint of each of these subintervals to the upper boundary of the property. These measurements give the values of f(x) at
x = 10,
30, 50, and 70. What is the approximate area of the lot?
