A person walking along a straight path has her velocity in miles per hour at time given by the function , for times in the interval . The graph of this function is also given in each of the three diagrams below.

Three ways to compute area.

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.