Calculus: Analytic Geometry and Calculus, with Vectors

260 Integrals

monotone increasing over the interval. This means that F(xi) < F(x2)

whenever a 5 x1 < x2 < b. Such functions F arise in many ways. We

can, for example, let P (a number) denote the population of an island,

state, or country, let P(x) denote the number of persons having age less

than or equal to x, and let

(4.71) F(x) =

P(x)

P

We can also suppose that the interval a 5 x < b represents a line segment

Figure 4.72

or a slim beam, as in Figure 4.72,

upon which sand and perhaps other

things are piled and from which hams

and other things are hung, and let

F(x) be the total mass which rests

upon or hangs from the part of the

interval from a to x. Because of the vividness of the latter interpretation,

F is sometimes called a mass function even when F(x) is a number which

is important in social sciences and which has nothing whatever to do with

such things as pounds and tons and grams and slugs. When x and

x + Ax both lie in the interval from a to b, the difference quotient

(4.73)

F(x + Ax) - F(x)

Ax

represents the average mass per unit length or the average linear density

over the interval with end points at x and x + Ax. In case this quotient

has, for a given x, a limit as Ax approaches zero, this limit is called the

density at x. When this density exists, we call it f(x) so that, by our

definition of derivatives, f(x) = F'(x). This idea of density has its

simplest applications in cases where F(x) has a continuous derivative.

In these cases the function f having values f(x) is called the density

function of the mass function F, and f(x) = F(x) for each x.

We are now ready to start introducing moments. Without assuming

that F is differentiable or even continuous, let t(xi) be a number (or point)

not necessarily in the interval from a to b, and let p be an integer which

is either 0 or positive. Let P be a partition of the interval a S x <_ b as

shown in Figure 4.74. The number F(xk) - F(xk_l) is the number

obtained by starting with the total mass in the interval a 5 x < xk and

Figure 4.74

xr 4

o t 1.

f a=x (^0) xl X2
XA*
-1-4-
X k-1
T
xh xA-