TABLE OF CONTENTS We considered tagged division of intervals in Section 12.5 of Chapter 12, particularly in Inequalities (12.15). Throughout this chapter T will be a set of dividing points of an interval [a, b]
and X the set of tags X k ?[ t k ?1, t k]. The symbol TX will denote the tagged division with dividing points T and tags X.
With a function f: [ a, b] 
and a tagged division TX we associate the sum
This sum is called a Riemann sum, or more explicitly the Riemann sum corresponding to the function f and the tagged division TX.
If no confusion can arise we shall abbreviate R(f, TX) to R(f) or R(TX) or even just R. We shall only do this if the objects left out in the abbreviated notation (like TX in R(f)) are fixed during the discussion.
The geometric meaning of R is indicated by Figure 15.1 for a non-negative function f. The sum R is simply the area of the rectangles which intersect the graph of f. Our intuition leads us to believe that, for a non-negative f, as a tagged division becomes finer and finer the sums R approach the area of the set {(x, y); a ?x ?b, 0 ? y ?f(x)}.
