Newton's Integrability Proof

f(x)

increasing

decreasing

discontinuous

sum

both

lower

upper

slide

subdivide

remove last point

reset

Upper-lower = ∑

= 2.13873 ≤ 3. = (max-min)Δx

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.