696 Iterated and multiple integrals

whenever the sum is a Riemann sum formed for the function f and for a

partition Q of S for which IQj < b, then f is said to be Riemann integrable

over S and I is said to be the Riemann integraloff over S. The integral

is usually denoted by the symbol

(13.622) fffsf(P) dS.

The integral is called a triple integral, and the three integral signs serve

to remind us that S is a three-dimensional set,that is, a set in E3 having

positive volume. As in previous cases, it is a convenience (and sometimes

also a source of confusion) to introduce the notation of limits and write

(13.623)

n

fff f(P)dS= lim

1Q1- O k=1

f(Pk) tSk

or

(13.624) f f fs f(P) dS = limj f(P) AS.

The following theorem, which is analogous to Theorem 13.38, is very

useful.

Theorem 13.63 If S is a subset of a region R consisting of points

(x,y,z) for which a, < x < a2, b1 < y < b2, c1 <_ z =< C2, if f(x,y,z) = 0

when (x,y,z) is a point in R but not in S, and if the eight integrals

11 = fffsf(x,y,z)

dS,

is =

Is =

I, =

f a dx

at f b dy f I f(x,y,z) dz,bt et

fbb,

dy faa dx fct f(x,y,z) dz,

f

ex

dz

a: dx f b,

e, Jul bif (x,y,z) dy,

all exist, then

12 =

fffff(x,y,z) dS

(^14) =faa= dxfo, dzfbb=f (x,y,z) dy
16 = fbb dy f , dz faa' f(x,y,z) dx
Is

f" dz

f

b= dyJ a f(x,y,z)

ct b, at dx

11=12=13=14=15=16=17=18,

that is, the eight integrals are all equal.

Remarks analogous to those following Theorem 13.38 apply here.

Proof of the theorem lies far beyond the scope of this course. We can

be content with a hazy understanding of the fact that the triple integrals

Il and 12 will exist if f is bounded and the set D of discontinuities of f

has volume (three-dimensional Lebesgue measure) 0. So far as ele-

mentary applications to elementary problems are concerned, we can be

sure that if the set S and the function f are bounded, then all of the integ-

rals appearing in the theorem must exist and must have the same value.

To develop a technique for setting up triple and iterated integrals,