Calculus: Analytic Geometry and Calculus, with Vectors

668 Iterated and multiple integrals

polar, for example) to determine a point P of S, we do not at present

allow any one brand of coordinates to dominate our work. We suppose

that we have a bounded function f defined over S and use the symbol

f(P) to denote the value of f at P. For example, if S is a lamina, f(P)
could be the density (mass per unit area) atP or the product of the density
atP and the specific heat atP and the temperature atP. If S is a lamina
and we want to calculate its moment of inertia about a line L, f(P) could
be the product of the density at P and the square of the distance from
P to L. It is often helpful to think of I f (P) I as being the height at P of a
solid which stands upon the base S.
The first step in our approach to a Riemann sum is to make a partition
Q (the letter P has been preempted) of the set S into n subsets S1, S2,
,Sn. As Figures 13.31 and 13.32 indicate, the result of partitioning

Figure 13.31 Figure 13.32

a set S in E2 into smaller subsets is not as simple as the result of parti-

tioning an interval in E1 into subintervals. The only things we require

of the sets S1j S2,. ,S. is that they be nonoverlapping, that their

union be S, and that each of them have positive area. It turns out that

the notational transition from Riemann sums to Riemann integrals will

be facilitated by denoting the areas of the sets S1, S2, -. ,S. by the

symbols AS,, AS2, ... ,ASn. The meanings of our symbols should be

emphasized. For each k = 1, 2, , n,the symbol ASk does not stand

for a part of the set S; it stands for the area of a part of the set S. For

each k = 1, 2, - - , n,let Pk be a point in the set Sk. The number RS

(Riemann sum) defined by

n

(13.33) RS = I AN ASk

k=1

is then a Riemann sum formed for the function f and for the partition Q

of S.

In order to tell what we mean by the norm jQJ of the partition Q, it is

necessary to introduce a simple geometrical concept. The diameter of a

set is the least upper bound of distances between pairs of points of a set.

The norm JQJ of the partition Q of S into subsets S1, S2,.. ,S. is the

greatest of the diameters of the sets S1, S2, ,Sn. We are now ready

to define the Riemann integral of f over S, the definition being analogous

to that involving (4.23). If there is a number I such that to each E > 0