Calculus: Analytic Geometry and Calculus, with Vectors

13.6 Triple integrals; rectangular coordinates 697

x=a1

x=a2

Y=gi(x)

z=f1(x>Y)

Y=g,(x)

Figure 13.64

we consider an example. As in Figure 13.64, let S be the setor solid

bounded by the surfaces having the equationsz = fl(x,y), z = f2(x,y),

y = gl(x), y = g2(x), x = a1, and x = a2. We suppose that, at each

point (x,y,z) of the solid, the solid has density (massper unit volume)

S(x,y,z). This means that to eache > 0 there corresponds a S > 0 such

that

(13.641) S(x,y,z) 1m < e

- 2T

whenever AS is the volume of a part of the solid containing the point
(x,y,z) and having diameter less than S, and Am is the mass of the part.
In case there is a constant k such that 6(x,y,z) = k whenever (x,y,z) is a
point of S, the solid is said to be homogeneous. While the ideas can be
applied in some other cases, we suppose that all of the functions which we
have introduced are continuous. Supposing that xo is a given number
and that p is a nonnegative integer that is 0 or 1 or 2 in most applications,
we set up integrals for M2p'xo, the pth moment of the solid about the plane
x = xo. Using rectangular coordinates, we slice S into subsets by planes

parallel to the coordinate planes. A typical subset, such as one shown
in Figure 13.64, has volume Ax Ay Az. Letting (x,y,z) be a point in the
subset, we use the number

(13.642) S(x,y,z) Ax Ay Az

(the product of mass per unit volume and volume) as an approximation
to the mass of the subset. If the total mass of the subset were concen-
trated at the point (x,y,z), its pth moment about the plane x = xo would
be

(13.643) (x - xo)PS(x,y,z) Ax Ay As.