Calculus: Analytic Geometry and Calculus, with Vectors

4.7 Mass, linear density, and moments 261

subtracting the total mass in the interval a < x < xk_l. Thus it is the
total mass in theinterval xk1 < x < xk. The number

(4.75) (xk - )P[F(xk)- F(xk-1)]

represents the pth momentabout the point of a single particle of mass

F(xk) - F(xk_1) concentrated at the point xk and, when the norm of P
is small, thisshould be a good approximation to the pth moment about
of the total mass in the interval xk_1 < x S xk. Moreover, the sum

(4.76)

n

Y (xk - )"[F(xk) - F(xk-1)]

km1

should be a good approximation to the pth moment about of the total

mass in the interval a5 x < b. Our statement about (4.76) was neces-

sarily vague and optimistic because the quantity we are trying to calculate

has not yet been defined. It is a fundamental fact, which is proved in

the theory of Riemann-Stieltjes integrals, that there is a number M`=P'e

such that the sum in (4.76) is near it whenever JPI is small, that is,

n

(4.77) Mgt = lim I (xk - )P[F(xk) - F(xk-1)]

IPI- 0 k=i

This number M??'F is called the pth moment about the point of the mass

in the interval a 5 x 5 b. In case p = 0, the pth moment is the total

mass in the interval a 5 x < b. In mechanics, the second moment is

called moment of inertia. In statistics and elsewhere, the particular

number x for whichMyi>-= 0 is called the mean (or mean value) of F over

the interval a < x 5 b. In mechanics and elsewhere, the point having

coordinate x is called the centroid of the mass. The number M"'==, the

second moment about the centroid or mean, is particularly important in

mechanics and statistics.

The above discussion applies equally well to mass functions F that

possess continuous density functions f and to those that do not. When

F does possess a continuous density function f, we can solve problems with

the aid of only Riemann integrals. In the latter case the number

(4.78) f (xk) Oxk

is taken to be an approximation to the total mass in the interval

xk-1 < x S Xk, and instead of (4.76) we use the Riemann sum

n

(4.781) 1 (xk - )f(xk) .xk

k=1

as an approximation to M"". Taking limits as the norm of the partitionX=t

P approaches 0 then gives the formula

(4.782) MZe't = f ' (x - )Pf(x) dx.