Calculus: Analytic Geometry and Calculus, with Vectors

13.3 Double integrals 669

there corresponds a S > 0 such that

(13.34)

nn

I-k 1 f (Pk) Ask I<

whenever the sum is a Riemann sum formed for the function f and fora
partition Q of S for which JQJ < S, then f is said to be Riemann integrable
over S and I is said to be the Riemann integral off over S. This integral
is usually denoted by the symbol

(13.35) f f, f(P) dS,

which displays the function f and the symbol S that represents theset

which was partitioned to obtain the approximating Riemann sums. The

integral is called a double integral because the set S is two-dimensional,

that is, a set in E2 having positive area. The two integral signs serve to

remind us that S is two-dimensional, but sometimes one of them is

omitted from the symbol. As was the case for simple (that is, one-

dimensional) integrals, it is a convenience (and sometimes also a source

of misunderstanding, confusion, and controversy) to drag in the nota-

tion of limits and write

(13.36) f f f (P) dS= lim Z f(Pk) OSk

IQI-O k=1

or

(13.37) f f, f(P) dS= limI f(P) AS.

When we are interested in problems in which a function f defined over
a bounded set S is involved and rectangular coordinates are to be used,
we can produce substantial simplifications of our work by letting R be a
rectangular set which contains the set S and by extending the domain of
f by putting f(x,y) = 0 when (x,y) is a point of R which is not in S. The
following theorem then enables us to evaluate double integrals by evalu-
ating iterated integrals.
Theorem 13.38 If S is a subset of the rectangular region R consisting
of points (x,y) for which a < x <_ b and c S y < d, if f(x,y) = 0 when
(x,y) is a point in R but not in S, and if the four integrals

(13.381) I3 = ff, ff(x,y) dS, I2 =f f,, f(x,y) dR,

(13.382) Ia=fb dx fdf(x,Y) dY, I, = f d dy f ab f(x,Y) dx

all exist, then

(13.383) I1=I2=13=14,

that is, the four integrals are all equal.