Calculus: Analytic Geometry and Calculus, with Vectors

13.4 Applications of doubleand iterated integrals

679

A simple modification of the precedingparagraph gives an iterated

integral in which the first integration is with respect tox.

Instead of

(13.441), we form the sum

(13.45)

Ay I (xv fixed - xo)PS(x,y) Ax,

where the part "y fixed" of the symbol informsus that the sum contains

only terms arising from subrectangles whichcomprise a horizontal strip

such as that shown in Figure 13.446. When the numbers Ax and 9tiv

are all small, (13.45) and

(13.451) Ay

i

(x - xo)PS(x,y) dx

are good approximations to the pth moment about theline x = xo of the

part of the lamina in one strip, and weare led to the formula

(13.452) Mzp'z0 = lim Ay j (xf1 - xo)PS(x,y) dx

and hence to the formula

(13.453) MiP'y, =

fo'dy

f,,_ (x-xo)PS(x,y)dx

for the pth moment about the linex = xo of the whole lamina.

Several quite simple and obvious remarkscan now be made. In order

to obtain derivations of formulas for M(P),,,, the pth moment of the lamina

about the line y = yo, it suffices to replace the factor (x - xo)P by the

factor (y - yo)P in the above derivations. In case p = 0, the factors

(x - xo)P and (y - yo)P are both equal to 1 and the numbers M=a),,

and M,')) are both equal to the mass M of the lamina. Thus

(13.454) M = f f, S(x,y) dx dy,

and we can replace this double integral by iterated integrals.
In case p = 1, the moments are first moments or moments of first order.
In case xo and yo are chosen such that MZ''s,= 0 and M,('.)., = 0, the
point (xo,yo) is called the centroid of the lamina. It is customary to let
x and y (x bar and y bar) denote the coordinates of the centroid. The
equations which determine x and y then become Ma)z = 0, My', = 0
or, as we see from (13.433) and the similar formula for Myl),,,,

(13.46) f fs (x - z)S(x,y) dx dy = 0, f f, (y- 3(x,y) dy = 0.

Since 9 and are constants that can be moved across integral signs, we