Calculus: Analytic Geometry and Calculus, with Vectors

706 Iterated and multiple integrals

nates p, ,, z is 3(p,4,z). Set up a threefold iterated integral in polar coordinates
for the total mass M of the solid. Ans.:

M _ f2Ad rs dp S z) dz + r2x do / N/ 1-0 p dp 10 6(x

/0 o p P L o a f o, dz.

10 As we near the end of our textbook, we can and should review and sum-

marize some of our ideas about integrals. This problem does not require us to

produce a specific numerical answer to a specific problem; it requires us to think

in general terms about methods by which such answers are produced. With the

understanding that the ideas have applications to more complicated situations

as well as to simpler ones, we consider the gravitational force F exerted upon a

particle of mass m at a point Q in E3 by a body B. This body B may be one-

dimensional, that is, it may be concentrated upon a one-dimensional set S which

may be a line segment or an arc of a curve having positive length. In this case

we suppose that the body has linear (or one-dimensional) density b(P) at the

point P in S. The body B may be two-dimensional, that is, it may be concen-

trated upon a two-dimensional set S which may be a circular disk or some other

region (on a plane or curved surface) which has positive area. In this case, we

suppose that the body has areal (or two-dimensional) density b(P) at the point

P in S. Finally, the body B may be an ordinary three-dimensional solid body,

that is, it may occupy a set S in E3 having positive volume. In this case we

suppose the body has ordinary (mass per unit volume) density S(P) at the point

P in S. We simplify and unify our discussion of these things by considering

length to be one-dimensional measure, area to be two-dimensional measure, and

volume to be three-dimensional measure. Thus we handle all of our examples

together by saying that we have a body B occupying an n-dimensional set S in

E3 having positive n-dimensional measure ISI and that the body has n-dimensional

density b(P) at the point P in S. The integer n may be 1 or 2 or 3. To start

the process of calculating F, we make a partition of the set S into q (note that n

has been preempted) measurable subsets S1, S2, - - , S. It would be thor-

oughly reasonable to denote the measures of these sets by IS1I, ISsI,- - - ,

IS5I, but we find it convenient to denote the measures by AS,, AS2, - -, AS,-

Thus, for each k, ASk is not a part of S; it is the measuret of a part of S. The

o which ASk is the measure. We use the number

AFFk S(Pk) OSk to approximate the mass of the part of the

Q body occupying the set S. For each k, we apply the

Figure 13.792 Newton inverse square law to obtain the force OFk

which a particle of mass S(Pk) ASk at Pk exerts upon

the particle of mass m at Q. Even small figures can be helpful, and we can

look at Figure 13.792. We find that

(1) AFk =Gm6(Pk) ASk QPk

IQPkI2 I QPk

f Perhaps we should recognize the fact that thevery useful standard notation is a relic

of the good old days when it was not the fashionto recognize a difference between a set and

the measure of the set.

next step is to select a point Pk (or Pk) in Sk. Note

-OPk that Pk is not in ASk but that Pk is a point in the set

f