Calculus: Analytic Geometry and Calculus, with Vectors

678 Iterated and multiple integrals

With the aid of Theorem 13.38 we can quickly express the above

double integral as an iterated integral in two different ways. It is,

however, worthwhile to learn to use a procedure which leads directly

to iterated integrals without making use of double integrals. As in the

preceding paragraph, we observe that Ax Ay is the area of a subrectangle

and use the number S(x,y) Ax Ay as an approximation to the mass of the

part of the lamina within the subrectangle. Again we note that if this

total mass were concentrated at the point (x,y), its pth moment about the

line x = xo would be

(13.44) (x - xo)DS(x,y) Ax Ay.

We then form the sum

(13.441) Ax I (x - xo)PS(x,y) Ay

x fixed

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

contains only terms arising from those subrectangles which comprise a

vertical strip such as that shown in Figure 13.445. When the numbers

Ax and Ay are all small, the coefficient of Ax in (13.441) is a Riemann sum

which is a good approximation to the coefficient of Ax in the expression

(13.442) dx fax (x - xo)Pb(x,y) dy.

Using this as an approximation to the pth moment about the line x = xo

of the part of the lamina in one strip, we are led to expect that the sum in

(13.443) M(p)g= lim I 0xfox= (x- xo)PS(x,y) dy

will be good approximation to the required total moment when the num-

bers Ax are all small and hence that (13.443) should be a valid formula.

This gives

(13.444) M?9_'=, = f of dx fox (x - xo)pS(x,y) dy

because the sum in (13.443) is a Riemann sum which approximates the

integral in (13.444).

Figure 13.445 Figure 13.446