Calculus: Analytic Geometry and Calculus, with Vectors

13.6 Triple integrals; rectangular coordinates 695

of agriculture are required to study analytic geometry and calculus so they can
start learning about these things.

13.6 Triple integrals; rectangular coordinates

titioned sets in El and E2 into subsets and used these partitions in the

process of setting up Riemann sums and Riemann integrals of functions.

One reason for the importance of what we have done lies in the fact that

the methods are easily extended to provide information about triple in-

tegrals of the form

(13.61) fffsf(P) dS

in which S is a set in three-dimensional space E3 in which it is sometimes

convenient to suppose that we exist.

Let S be a bounded set in E3 which may be a spherical ball (the set

consisting of the points inside and the points on a sphere) or any other

bounded set in E3 which has a positive volume ISI. As was the case

when we defined double integrals, we do not allow any one brand of

coordinates to dominate our work. We suppose that we have a bounded

function f defined over S and use the symbol f(P) to denote the value

of f atP. For example, f(P) could be the density (mass per unit volume)

at P or the product of the density at P and the specificheat at P and the

temperature at P. The first step in our approach to a Riemann sum is

to make a partition Q (again the letter P has beenpreempted) of the set

S into n subsets S1, S2,. ,S,,. The only thing we require of the

sets Si, S2, ... , Sn is that they be nonoverlapping,that their union

be S, and that each one of them have positive volume. The notational

transition from Riemann sums to Riemann integrals is facilitated by

denoting the volumes of the sets S1, S2, ,Sn by the symbols AS1,

AS 2j ,AS.. Thus, for each k = 1, 2, , n,the symbol OSk does

not stand for a part of the set S; it standsfor the volume of a part of S.

For each k = 1, 2, , n,let Pk be a point in the set Sk. The number

RS (Riemann sum) defined by

(13.62)

n

RS = I f (Pk) O Sk

ks1

is then a Riemann sum formed for the function fand for the partition

Q of S. The norm JQI of the partition Q is, as inprevious cases, the

greatest of the diameters of the subsets. If there is a number I such that

to each e > 0 there corresponds a a >0 such that

n

(13.621) II- 1 f(Pk)1SkI<e

k=1