Lower and Upper Sums

The area problem asks how we can determine the area under a non-straight curve. It turns out that we can approximate the area using rectangles. The more rectangles we use, the better our approximation to the area is. The applet below allows you to use different numbers of rectangles to approximate the area in two ways. The exact area (to four decimal places) is given in the middle and you can choose whether or not to consider lower sums, upper sums, or both. Each method gives an approximate area using rectangles (move the slider to change ). The difference between the exact answer and the approximate answer is given as the error of each approximation -- the lower sum is always less than the actual area while the upper sum is always greater than the actual area. The value for is the width of each rectangle. We will take this idea and consider the limit as to find the exact area. Note that as .