Upper and lower sums for increasing functionsa. Suppose

Chapter 5, Problem 83E

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Upper and lower sums for increasing functionsa. Suppose the graph of a continuous function ƒ(x) rises steadily as x moves from left to right across an interval [a, b]. Let P be a partition of [a, b] into n subintervals of length ?x = (b - a)/n. Show by referring to the accompanying figure that the difference between the upper and lower sums for ƒ on this partition can be represented graphically as the area of a rectangle R whose dimensions are [ƒ(b) - ƒ(a)] by ?x. (Hint: The difference U – L is the sum of areas of rectangles whose diagonals Q0Q1, Q1Q2, …, Qn-1Qn lie along the curve. There is no overlapping when these rectangles are shifted horizontally onto R.)b. Suppose that instead of being equal, the lengths ?xk of the subintervals of the partition of [a, b] vary in size. Show that where ?xmax is the norm of P, and hence that (U – L) = 0.