Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MULTIPLE INTEGRALS

z

c

dx

a

x

dz

dy

b y

dV=dx dy dz

Figure 6.3 The tetrahedron bounded by the coordinate surfaces and the

planex/a+y/b+z/c= 1 is divided up into vertical slabs, the slabs into

columns and the columns into small boxes.

Find the volume of the tetrahedron bounded by the three coordinate surfacesx=0,y=0

andz=0and the planex/a+y/b+z/c=1.

Referring to figure 6.3, the elemental volume of the shaded region is given bydV=zdxdy,
and we must integrate over the triangular regionRin thexy-plane whose sides arex=0,
y=0andy=b−bx/a. The total volume of the tetrahedron is therefore given by

V=

∫∫

R

zdxdy=

∫a

0

dx

∫b−bx/a

0

dy c

(

1 −

y

b

−

x

a

)

=c

∫a

0

dx

[

y−

y^2

2 b

−

xy

a

]y=b−bx/a

y=0

=c

∫a

0

dx

(

bx^2

2 a^2

−

bx

a

+

b

2

)

=

abc

6

.

Alternatively, we can write the volume of a three-dimensional regionRas

V=

∫

R

dV=

∫∫∫

R

dx dy dz, (6.7)

where the only difficulty occurs in setting the correct limits on each of the

integrals. For the above example, writing the volume in this way corresponds to

dividing the tetrahedron into elemental boxes of volumedx dy dz(as shown in

figure 6.3); integration overzthen adds up the boxes to form the shaded column

in the figure. The limits of integration arez=0toz=c

(

1 −y/b−x/a

)

,and