January 2014
Intermediate to advanced
144 pages
4h 2m
English
Content preview from Mathematical Formulas for Industrial and Mechanical Engineering
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5.35 Triple Integrals
A triple integr al has a three-dimensional doma in. Since the resulting function exists
in four dimensions, the function cannot be represented graphically. However, some
mathematicians label the results of a triple integral as hypervolume. The formula
for calculating a triple integral can be determined as
lim
m!N
lim
n!N
lim
p!N
X
m
i51
X
n
j51
X
p
k51
f ðx
ijk
; y
ijk
; z
ijk
ÞΔz
jk
Δy
jk
Δx
jk
5
ððð
E
f ðx; y; zÞdV
5
ð
b
a
ð
d
c
ð
f
e
f ðx; y; zÞdz dy dx
Example
Use a triple integral to find the volume of the solid bounded by the graphs of z 5 x
2
1 y
2
and the plane z 5 4.
Solution
The following graph shows a plot of the paraboloid z 5 x
2
1 y
2
(in blue), the plane z
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