This figure is based on Newton's proof of the integrability of monotonic functions found in his Principia Mathematica (Book I, Lemma III). The error between the lower and upper sums, represented by the yellow rectangles, slides over and fits in a rectangle whose height is the height of the graph and width is that of the broadest yellow rectangle. As the partition is subdivided, the error approaches zero. In other words, the upper and lower sums approach the same value, the value of the integral of the function.
Click to add a point to the partition or click the subdivide button to divide each interval in half. The "slide" slider shows that the difference between the upper and lower sums fits inside a rectangle whose height is the difference in the values of at the ends of the interval and whose base is the width of the widest interval of the partition.
Snapshots 1 and 2: the decreasing function with the partition subdivided a couple of times, with the (yellow) difference rectangles in their original position and slid over inside the outlined rectangle, respectively
Snapshot 3: the increasing function with the (yellow) difference rectangles slid over inside the outlined rectangle
Snapshot 4: the discontinuous function with the (yellow) difference rectangles slid over inside the outlined rectangle