(^670) Iterated and multiple integrals
Proof of this theorem lies far beyond the scope of this course. Persons
who continue study of mathematical analysis until theories of Lebesgue
measure and Lebesgue integrals (including a theorem known as the Fubini
theorem) have been learned will find that validity of the theorem will be
an easy consequence of fundamental relations between Riemann and
Lebesgue integrals. For the present, we can be content with a hazy
understanding of the fundamental fact that the double integrals Ir and
12 will exist if f is bounded and the set D of discontinuities off has area
(two-dimensional Lebesgue measure) 0 and, moreover, the iterated
integrals 13 and 14 will also exist if it is also true that each horizontal line
and each vertical line intersects D in a set having length (one-dimensional
Lebesgue measure) 0. So far as elementary applications to elementary
problems are concerned, we can be sure that if the set S and the function f
are bounded, then the double integrals in (13.381) and the iterated
integrals in (13.382) must exist and must have the same value.
Symbols used for iterated integrals were discussed in Section 13.1.
In addition to the symbols used in this section for double integrals,
those appearing in the formula
(13.384) ffsf(x,y) dS = ffs f(x,y) dx dy = fs f(x,y) dx dy
are sometimes used when rectangular coordinates are involved.
Problems 13.39
1 Let S be the set in E2 bounded by the graphs of the equations
Y=x2, Y=x+2.
Supposing that f is continuous over S and that
f =ffs f(x,Y) dS,
sketch a figure which displays the set S and an appropriate rectangular region R
and then write complete and intelligible descriptions of the steps involved in
using Theorem 13.38 to obtain the formulas
J _ f21 dx fx=+2f(x,Y)
dy
and
- for dy f, f(x,Y) dx -I-
f,4
dyfvVI-V
2 f(x,Y) dx.(^2) Supposing that 0 < a < b and that f is continuous, determine a set S
in the xy plane such that
f a dy
fvv+b f(x,Y)
dx
=ffsf(x,y) dS.