282 Chapter 9Functions of several variables
This is called a double integraland is evaluated by integrating first with respect to one
variable and then with respect to the other. The value of the integral does not depend
on the order in which the integrations are performed if the function is continuous
within the ranges of integration, but some care must be taken when discontinuities
are present.
EXAMPLE 9.29Evaluate the integral (9.53) whenf(x, y) 1 = 12 xy 1 + 13 y
2.
(i) Integrating first with respect to x, using the result of Example 9.28(i),
(ii) Integrating first with respect to y, using the result of Example 9.28(ii),
0 Exercises 64, 65
When the order of integration is not given explicitly, some care must be taken to
associate each pair of integration limits with the appropriate variable. Thus, if a triple
integral is written as
the conventionis that the integration is to be carried out from the inside outward, with
(a,b)the limits for x,(c,d)for y, and(e,f)for z:
The concepts and methods of integration discussed in Chapters 5 and 6 for integrals
over one variable apply to multiple integrals, with some changes in interpretation.
The double integral is discussed in some detail in the following sections. The special
case of the triple integral and its importance for the description of physical systems in
three dimensions are discussed in Chapter 10.
Ifxyzdxdydz
efcdab=,,
ZZZ()
I f x y z dxdydz
efcdab=,ZZZ(),
=− −+−−
1
2
2222 33()()()()dcba dcba
ZZ Z
abcdab() ()()23xy y dy dx x d c d c
22233+=−+−
dx
=− −+−−
1
2
2222 33()()()()dcba dcba
ZZ Z
cdabcd() ()()23xy y dx dy y b a 3 y b a
2222+=−+−
dy