Mathematical Methods for Physics and Engineering : A Comprehensive Guide

6.3 APPLICATIONS OF MULTIPLE INTEGRALS

and letN→∞as each of the volumes ∆Vp→0. If the sumStends to a unique

limit,I, then this is called thetriple integral off(x, y, z)over the regionRand is

written

I=

∫

R

f(x, y, z)dV , (6.5)

wheredVstands for the element of volume. By choosing the subregions to be

small cuboids, each of volume ∆V=∆x∆y∆z, and proceeding to the limit, we

canalsowritetheintegralas

I=

∫∫∫

R

f(x, y, z)dx dy dz, (6.6)

where we have written out the element of volume explicitly as the product of the

three coordinate differentials. Extending the notation used for double integrals,

we may write triple integrals as three iterated integrals, for example,

I=

∫x 2

x 1

dx

∫y 2 (x)

y 1 (x)

dy

∫z 2 (x,y)

z 1 (x,y)

dz f(x, y, z),

where the limits on each of the integrals describe the values thatx,yandztake

on the boundary of the regionR. As for double integrals, in most cases the order

of integration does not affect the value of the integral.

We can extend these ideas to define multiple integrals of higher dimensionality

in a similar way.

6.3 Applications of multiple integrals

Multiple integrals have many uses in the physical sciences, since there are numer-

ous physical quantities which can be written in terms of them. We now discuss a

few of the more common examples.

6.3.1 Areas and volumes

Multiple integrals are often used in finding areas and volumes. For example, the

integral

A=

∫

R

dA=

∫∫

R

dx dy

is simply equal to the area of the regionR. Similarly, if we consider the surface

z=f(x, y) in three-dimensional Cartesian coordinates then the volume under this

surface that stands vertically above the regionRis given by the integral

V=

∫

R

zdA=

∫∫

R

f(x, y)dx dy,

where volumes above thexy-plane are counted as positive, and those below as

negative.