13.4 Applications of doubleand iterated integrals
677
tions y = 0, y = x2, and x = 1. We suppose that, at each point (x,y)
of the lamina, the lamina has areal density (ormass per unit area) S(x,y).
This means that to each e > 0 there correspondsa S > 0 such that(13.42) S(x,Y)- OSI < e
whenever AS is the area of a part of the lamina containing the point
(x,y) and having diameter less than S andAm is the mass of the part.
In the simplest applications, there isa constant k, which may be 1, such
that S(x,y) = k whenever (x,y) is a point of theset S occupied by the
lamina; in this case the lamina is saidto be homogeneous. While the
ideas can be applied in some other cases,we suppose that S is continuous.
Supposing that xo is a given number and thatp is a given nonnegative
integer that is 0 or 1 or 2 in most applications,we undertake to learn the
techniques involved in setting up three differentexpressions for Mzm),,,
the pth moment of the lamina about the linex = xo.
To set up a double integral for M"" 21, we chop the rectangle of Figure
13.41 into subrectangles by lines parallel to the coordinate axes. A
particular subrectangle, such as theone shown in Figure 13.41, has area
Ax Ay. Supposing that the subrectangle lies entirely withinthe lamina,
we select a point (x,y) in the 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. If this total mass were concentratedat the point (x,y), its
pth moment about the line x = xo would be(13.43) (x - xo)PS(x,Y) Ax Ay.We therefore use this number as an approximation to the pthmoment
about the line x = xo of the part of the lamina in theone subrectangle.
The sum(13.431) T'(x - xo)PS(x,y) Ax Ay,which contains a term for each subrectangle in the lamina, should then
be a good approximation to the total pth moment of the entire lamina
whenever the diameters of the subrectangles are all small. This leads us
to the formula(13.432) M x0 =lim E(x - xo)PS(x,y) Ax Ay,the right side of which is taken to be the definition of the number M"'=
which we are seeking. In accordance with the theory of double integrals
involving (13.34) and Theorem 13.38, the right side of (13.432) is a
double integral which we can denote by one or the other of the symbols
in the formula