Double integrals

6
Multiple integrals
For functions of several variables, just as we may consider derivatives with respect
to two or more of them, so may the integral of the function with respect to more
than one variable be formed. The formal definitions of such multiple integrals are
extensions of that for a single variable, discussed in chapter 2. We first discuss
double and triple integrals and illustrate some of their applications. We then
consider changing the variables in multiple integrals and discuss some general
properties of Jacobians.
6.1 Double integrals
For an integral involving two variables – a double integral – we have a function,
f(x, y) say, to be integrated with respect to x and y between certain limits. These
limits can usually be represented by a closed curve C bounding a region R in the
xy-plane. Following the discussion of single integrals given in chapter 2, let us
divide the region R into N subregions ∆Rp of area ∆Ap , p = 1, 2, . . . , N, and let
(xp , yp ) be any point in subregion ∆Rp . Now consider the sum
S=
N
f(xp , yp )∆Ap ,
p=1
and let N → ∞ as each of the areas ∆Ap → 0. If the sum S tends to a unique
limit, I, then this is called the double integral of f(x, y) over the region R and is
written
f(x, y) dA,
(6.1)
I=
R
where dA stands for the element of area in the xy-plane. By choosing the
subregions to be small rectangles each of area ∆A = ∆x∆y, and letting both ∆x
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MULTIPLE INTEGRALS
y
V
d
dy
dx
dA = dxdy
U
R
S
C
c
T
a
b
x
Figure 6.1 A simple curve C in the xy-plane, enclosing a region R.
and ∆y → 0, we can also write the integral as
I=
f(x, y) dx dy,
(6.2)
R
where we have written out the element of area explicitly as the product of the
two coordinate differentials (see figure 6.1).
Some authors use a single integration symbol whatever the dimension of the
integral; others use as many symbols as the dimension. In different circumstances
both have their advantages. We will adopt the convention used in (6.1) and (6.2),
that as many integration symbols will be used as differentials explicitly written.
The form (6.2) gives us a clue as to how we may proceed in the evaluation
of a double integral. Referring to figure 6.1, the limits on the integration may
be written as an equation c(x, y) = 0 giving the boundary curve C. However, an
explicit statement of the limits can be written in two distinct ways.
One way of evaluating the integral is first to sum up the contributions from
the small rectangular elemental areas in a horizontal strip of width dy (as shown
in the figure) and then to combine the contributions of these horizontal strips to
cover the region R. In this case, we write
y=d x=x2 (y)
f(x, y) dx dy,
(6.3)
I=
y=c
x=x1 (y)
where x = x1 (y) and x = x2 (y) are the equations of the curves T SV and T UV
respectively. This expression indicates that first f(x, y) is to be integrated with
respect to x (treating y as a constant) between the values x = x1 (y) and x = x2 (y)
and then the result, considered as a function of y, is to be integrated between the
limits y = c and y = d. Thus the double integral is evaluated by expressing it in
terms of two single integrals called iterated (or repeated) integrals.
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6.1 DOUBLE INTEGRALS
An alternative way of evaluating the integral, however, is first to sum up the
contributions from the elemental rectangles arranged into vertical strips and then
to combine these vertical strips to cover the region R. We then write
x=b y=y2 (x)
f(x, y) dy dx,
(6.4)
I=
x=a
y=y1 (x)
where y = y1 (x) and y = y2 (x) are the equations of the curves ST U and SV U
respectively. In going to (6.4) from (6.3), we have essentially interchanged the
order of integration.
In the discussion above we assumed that the curve C was such that any line
parallel to either the x- or y-axis intersected C at most twice. In general, provided
f(x, y) is continuous everywhere in R and the boundary curve C has this simple
shape, the same result is obtained irrespective of the order of integration. In cases
where the region R has a more complicated shape, it can usually be subdivided
into smaller simpler regions R1 , R2 etc. that satisfy this criterion. The double
integral over R is then merely the sum of the double integrals over the subregions.
Evaluate the double integral
x2 y dx dy,
I=
R
where R is the triangular area bounded by the lines x = 0, y = 0 and x + y = 1. Reverse
the order of integration and demonstrate that the same result is obtained.
The area of integration is shown in figure 6.2. Suppose we choose to carry out the
integration with respect to y first. With x fixed, the range of y is 0 to 1 − x. We can
therefore write
x=1 y=1−x
I=
x2 y dy dx
x=0
x=1
=
x=0
y=0
x2 y 2
2
y=1−x
1
dx =
0
y=0
x2 (1 − x)2
1
dx =
.
2
60
Alternatively, we may choose to perform the integration with respect to x first. With y
fixed, the range of x is 0 to 1 − y, so we have
y=1 x=1−y
I=
x2 y dx dy
y=0
y=1
=
y=0
x=0
x3 y
3
x=1−y
1
dx =
0
x=0
(1 − y)3 y
1
dy =
.
3
60
As expected, we obtain the same result irrespective of the order of integration. We may avoid the use of braces in expressions such as (6.3) and (6.4) by writing
(6.4), for example, as
b y2 (x)
dx
dy f(x, y),
I=
a
y1 (x)
where it is understood that each integral symbol acts on everything to its right,
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