Calculus: Analytic Geometry and Calculus, with Vectors

654 Iterated and multiple integrals

insist on the other hand that we should "work from right to left,"so

that f

c

goes with dy andJAgoes with dx. Of course, we could lengthen

a long story by insisting that we should always keep the parentheses and

avoid the tempests and the stories, but this is impractical. While we

reserve the option of using parentheses whenever we wish to do so, we

ordinarily remove parentheses and ambiguities from iterated integrals

by writing the integrals in such a way that each integral sign except the

one on the right is immediately followed by the symbol showing the

variable of integration which has the limits of integration appearing on

the integral sign. Thus, for example,

(13.17)4

f2d f 2d=

f2[3]::4

JI y=3

_ 3 (^37) J2xdx =--37X2]--2= 111
1 3 2 x31
6
x2 2 x=4
(13.18)
2
1 dy
14
xy2 dx =
f2
dy[ 2
x=3
7 2
=2 1 y2dy
and
7y32 49
2 31- 6
(13.181) I x dt f' (t- u) du =f xdtr-(t 2 u)2]u=t
f L U-0
x
t2 di -1 t3 t=x
xs
0 3]t-0 6
Note that, in each case, the integral appearing on the right is evaluated
first. Note also that when we are in the process of integrating with
respect to a particular variable, all other variables are temporarily con-
sidered to be constants. Opportunities to become familiar with these
things are provided by the following problems.

Problems 13.19

1 Show that

(a) Jol dx fox (x2 + Y.) dy= $

(c) fol dx Jox

dY f - ds =

g

(b) f ' di Jot (t^ + u") du=

n -{- 1

(n> -1)

(d)

U dx

Jxx+1

e-- dy = 1 - e 1

2 By evaluating all of the integrals involved, show that

(^12211221)
(a) o dx o x dy = Ja dy f0 xdx (b) Jo dx Jl x dy = Jl dy Jo xdx