082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

27.7 Some simple volume and surface integrals 889
27.7 SOMESIMPLEVOLUMEANDSURFACEINTEGRALS
Volume and surface integrals arise frequently in electromagnetism and fluid mechanics.
We will illustrate these concepts through some simple examples. For a thorough
treatment ofmore general cases you will need torefer toamore advanced text.
Engineeringapplication27.3
Themassofasolidobject
Consider a solid object such as that shown in Figure 27.17. The density, ρ, of this
objectmayvaryfrompointtopoint.So,atanypointPwithcoordinates (x,y,z),the
density is a function of position, that is ρ = ρ(x,y,z). Since density is a scalar, this
isan example ofascalar field, like those discussed inSection 7.4.
Supposeweselectaverysmallpieceofthisobjecthavingvolume δV andlocated
atP(x,y,z). Recall fromelementary physics that
density = mass
volume
Then the mass of thissmallpiece, δm, isgiven by
δm=ρδV
If we wish to calculate the total mass,M, of the object we must sum all such contributionsfromtheentirevolume.Thisisfoundbyintegratingthroughout
thevolume.
We write thisas ∫
total mass,M = ρdV
V
This is an example of a volume integral, so called because the integration is performed
throughout the volume. It will usually take the form of a triple integral such
as those discussed in Section 27.6.2. Technically, there are three integral signs, but
forbrevitythesehavebeenreplacedbythesingle ∫ V whereitistobeunderstoodthat
the integral is to be performed over a volume. In any specific problem care must be
takentoensurethattheentirevolumeisincludedwhentheintegrationisperformed.
For example, consider the case of a solid cube with sides of length 1 unit. Let
one corner be positioned at the origin and let the edges coincide with the positive
x, y and z axes. Suppose the density of the cube varies from point to point, and is
∫givenby ρ(x,y,z) =x+y+z.Thentheintegralwhichgivesthemassofthecubeis
(x + y + z)dV, where the volume V is the region occupied by the cube. This
V
integral has been evaluated in Example 27.16 and found to be 3 , representing the
2
mass of the cube.
volume V
volume dV
densityr(x, y,z)
P (x, y,z)
Figure27.17
Thetotal massofabodyis foundby
summing,orintegrating,throughout
the wholevolume.