284 Chapter 9Functions of several variables
that is, the volume between the representative surface of the functionf(x,y)and the
xy-plane.
9Alternatively, the integral can be interpreted as a property of the rectangular
region in the xy-plane. For example, iff(x,y)is a surface mass density, the differential
mass in element of areadA 1 = 1 dx 1 dyisdm 1 = 1 f(x,y)dAand the total mass is
(9.57)
where means integration over the (rectangular) region. In the special case
off(x,y) 1 = 11 , the integral is the area of the rectangle:
More generally, the region in the xy-plane need not be rectangular. Letf(x,y)be
continuous within and on a boundary of a region R of the xy-plane.
The form of the integral over the region depends on how the boundary is defined. In
Figure 9.13a the boundary is made up of two sections:y 1 = 1 g(x)‘below’ the extreme
pointsx 1 = 1 aandx 1 = 1 b,andy 1 = 1 h(x)‘above’ these points. Then, within R,
g(x) 1 ≤ 1 y 1 ≤ 1 h(x)fora 1 ≤ 1 x 1 ≤ 1 b
and the integral over the region (the sum of vertical strips) is
(9.58a)
In Figure 9.13b the boundary is made up of the two sections:x 1 = 1 p(y)to the ‘left’ of
the extreme pointsy 1 = 1 candy 1 = 1 d,andx 1 = 1 q(y)to the ‘right’. Then, within R,
p(y) 1 ≤ 1 x 1 ≤ 1 q(y)forc 1 ≤ 1 y 1 ≤ 1 d
IfxydA fxydy
abgxhx=,= ,
ZZZ
R() ()
()()ddx
ZZZ ZZ
RdA dx dy dx dy b a d c
cdababcd== =−−()()
Z
R⋅⋅⋅dA
MfxydA fxydxdy
cdab=,=ZZZ,
R() ()
9The double integral as a volume was discussed by Clairaut in his Recherchesof 1731. The double integral sign
notation (∫∫) was first used by Lagrange in 1760.
y
x
dx
y=h(x)
y=g(x)
R
a
b
..............................................................................................................................................................................................................................
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
....................................................................................................................................................................................................................................................................................
.......
........
.........
............
..............
........................
.........................................
...........................
................
...............
............
..........
..........
.........
........
.........
.......
.......
.......
.......
......
.......
.....
......
......
.....
.....
......
.....
.....
.....
....
.....
.....
....
....
.....
....
....
.....
....
....
....
....
....
....
....
....
....
....
....
.....
....
....
.....
....
....
......
.....
.....
......
......
......
........
.......
.........
...........
..............
....................
........................................................
......................
................
..............
............
..........
..........
.........
........
........
.......
.......
........
......
......
.......
.....
......
......
.....
.....
......
.....
....
......
....
.....
.....
....
....
.....
....
....
....
....
....
.....
....
...
.....
....
....
....
....
....
....
....
....
.....
.....
....
.....
.....
......
......
......
...
(a)
x=p(y)
x=q(y)
y
x
dy
R
c
d
.............................................................................................................................................................................................................................................................................................................
.........................
..........................
...........................
............................
.............................
.............................
..............................
...............................
.............................................................................................................................................................................................................................................................................................................................................
.......
........
.........
............
..............
........................
.........................................
...........................
................
...............
............
..........
..........
.........
........
.........
.......
.......
.......
.......
......
.......
.....
......
......
.....
.....
......
.....
.....
.....
....
.....
.....
....
....
.....
....
....
.....
....
....
....
....
....
....
....
....
....
....
....
.....
....
....
.....
....
....
......
.....
.....
......
......
......
........
.......
.........
...........
..............
....................
........................................................
......................
................
..............
............
..........
..........
.........
........
........
.......
.......
........
......
......
.......
.....
......
......
.....
.....
......
.....
....
......
....
.....
.....
....
....
.....
....
....
....
....
....
.....
....
...
.....
....
....
....
....
....
....
....
....
.....
.....
....
.....
.....
......
......
......
...
(b)
Figure 9.13