Calculus: Analytic Geometry and Calculus, with Vectors

698 Iterated and multiple integrals

We therefore use this number as an approximation to the pth moment

of the subset. The sum

(13.644) Z(x - xo)PS(x,y,z) Ax Ay Az,

which contains a term for each subset, should then be a good approxima-

tion to the total pth moment of the whole solid whenever the diameters

of the subsets are all small. This leads us to the formula

(13.645) M = lim E(x - xo)'S(x,y,z) Ax Ay Az,

the right side of which is taken to be the definition of the number M=P'zp

which we are seeking. In accordance with the definition of triple integ-

rals, the right side of (13.645) is a triple integral which we can denote by

one or the other of the symbols in the formula

(13.646) Mz-'Z, = fffs (x-xo)PS(x,y,z) dS

=f f fs (x- xo)"5(x,y,z) dx dy dz.

With the aid of Theorem 13.63, we can undertake to express the triple

integral in terms of iterated integrals in various ways. It is, however,

worthwhile to learn to use a procedure which leads directly to iterated

integrals. As above, we build up the expression

(13.65) (x - xo)'S(x,y,z) Ax Ay Az

to serve as an approximation to the required moment of a single subset.

We then form the sum

(13.651) Ax Ay I (x - xo)'S(x,y,z) Az,

x,11 fixed

where the part "x, y, fixed" of the symbol serves to inform us that the

sum contains only terms arising from those subsets which comprise a

single vertical column such as that shown in Figure 13.64. Thus (13.651)

and the number

I:(x.v)

(13.652) dx Dy f

(x y) (x - xo)'S(x,y,z) dz

are approximations to the required moment of one column. Next we

form the sum

(13.653) Ax I Ay (x -

x fixed

Ji(x,y) xo)'S(x,y,z) dz,

where the part "x fixed" of the symbol serves to inform us that the sum

contains only terms arising from columns that comprise a slab running

from the cylinder on which y = yl(x) to the cylinder on whichy = y2(x).