Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MULTIPLE INTEGRALS

V

U

C

T

S

dx

dy

R

dA=dxdy

y

d

c

a b x

Figure 6.1 A simple curveCin thexy-plane, enclosing a regionR.

and ∆y→0, we can also write the integral as

I=

∫∫

R

f(x, y)dx dy, (6.2)

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 differentialsexplicitlywritten.

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

bewrittenasanequationc(x, y) = 0 giving the boundary curveC. 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 widthdy(as shown

in the figure) and then to combine the contributions of these horizontal strips to

cover the regionR. In this case, we write

I=

∫y=d

y=c

{∫x=x 2 (y)

x=x 1 (y)

f(x, y)dx

}

dy, (6.3)

wherex=x 1 (y)andx=x 2 (y) are the equations of the curvesTSVandTUV

respectively. This expression indicates that firstf(x, y)istobeintegratedwith

respect tox(treatingyas a constant) between the valuesx=x 1 (y)andx=x 2 (y)

and then the result, considered as a function ofy, is to be integrated between the

limitsy=candy=d. Thus the double integral is evaluated by expressing it in

terms of two single integrals callediterated(orrepeated) integrals.