Note that in each diagram, we use four rectangles to estimate the area under on the interval , but the method by which the four rectangles' respective heights are decided varies among the three individual graphs.
Think about how the heights of the rectangles in the left-most diagram are being chosen. Determine the value of by evaluating the function at appropriately-chosen values and observing the width of each rectangle. Note, for example, that .
Use the rectangles in the middle diagram to find the value of .
Use the rectangles in the right-most diagram to find the value of .
Which estimate do you think is the best approximation of , the total distance the person traveled on ? Why?
In order to get credit for this problem all answers must be correct.