Calculus: Analytic Geometry and Calculus, with Vectors

272 Integrals

fully rigorous about the matter, we assume that the Riemann integral in

(4.85) Y = f

b

1(x) dx

exists and is the volume Y of the set S. Our next step is to suppose that
the points (x,y,z) of the set S are points of a material body B such that,
for some positive constant S, a part of the body has mass 6 AV if that part
occupies a part of the set S having volume V. Our body B (which
really is somewhat different from the conglomeration of atomic particles
that constitute a potato) is said to be homogeneous because its density
(mass per unit volume) has the same constant value S at all places in the
body.
child thinks a potato is.
Supposing that is a number and
that p is 0 or a positive integer, we
should now find it easy to construct
formulas for calculation of the num-
xk
ax- V ber MZj E, the pth moment of the body
B about the plane having the equation
Figure 4.86 x =. Realizing that schematic
figures can be helpful even when
some wise people consider them to be semisuperfluous, we sketch Figure
4.86. We make a partition P of the interval a < x < b into subintervals.
Supposing as usual that xk_1 < xk < xk, we use the number
(4.87)

At last we have a body B which might, for example, be what a

A' (xk) AXk

as an approximation to the volume of the slab which lies between the

planes having the equations x = xk_i and x = xk. Multiplying by the

density S gives an approximation to the mass of the slab. When the

norm of P is small, all points of the slab lie at about the same distance

14 - kj from the plane having the equation x = t and multiplying the

mass by (xt* - k,)P should therefore give a good approximation to the pth

moment of the slab about the plane having the equation x =. The

Riemann sum

(4.871) S2(xk - )P.4(xk) AXk

should then be a good approximation to the moment of the whole body.

Since the Riemann sums have a limit which is the Riemann integral in

the right member of the formula

(4.872) Mw = S f ab (x - i;) P11(x) dx,

our work motivates the definition by which the required moment is

defined by this formula. Analogous formulas define moments about

planes parallel to the other coordinate planes.