Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MULTIPLE INTEGRALS

y

1

1

dy

0

0

dx x

x+y=1

R

Figure 6.2 The triangular region whose sides are the axesx=0,y=0and

the linex+y=1.

and that the order of integration is from right to left. So, in this example, the

integrandf(x, y) is first to be integrated with respect toyand then with respect

tox. With the double integral expressed in this way, we will no longer write the

independent variables explicitly in the limits of integration, since the differential

of the variable with respect to which we are integrating is always adjacent to the

relevant integral sign.

Using the order of integration in (6.3), we could also write the double integral as

I=

∫d

c

dy

∫x 2 (y)

x 1 (y)

dx f(x, y).

Occasionally, however, interchange of the order of integration in a double integral

is not permissible, as it yields a different result. For example, difficulties might

arise if the regionRwere unbounded with some of the limits infinite, though in

many cases involving infinite limits the same result is obtained whichever order

of integration is used. Difficulties can also occur if the integrandf(x, y) has any

discontinuities in the regionRor on its boundaryC.

6.2 Triple integrals

The above discussion for double integrals can easily be extended to triple integrals.

Consider the functionf(x, y, z) defined in a closed three-dimensional regionR.

Proceeding as we did for double integrals, let us divide the regionRintoN

subregions ∆Rpof volume ∆Vp,p=1, 2 ,...,N,andlet(xp,yp,zp) be any point in

the subregion ∆Rp. Now we form the sum

S=

∑N

p=1

f(xp,yp,zp)∆Vp,