Triple Integrals
John E. Gilbert, Heather Van Ligten, and Benni Goetz
Once we’ve seen the pattern of how to go from integrals of functions of one variable to double
integrals of functions of two variables, then studying triple integrals
E
f
(
x, y, z
)
dxdydz
of functions
f
(
x, y, z
)
of three variables is a natural step, and a very important one in applications.
Now
•
the domain of integration is a solid
E
in 3space, and
•
when
f >
0
the integral is interpreted as the volume of the solid in 4space below the graph of
f
and above
E
.
Examples show how to write the triple integral as a repeated integral:
Example 1:
express the triple integral
I
1
=
E
f
(
x, y, z
)
dxdydz
as a repeated integral when
E
is the solid shown to
the right above the rectangle
D
= [
a, b
]
×
[
c, d
]
in the
xy
plane and below the graph of
z
= 6

x
.
Think of
D
as the
base
of
E
and the graph of
z
= 6

x
above
D
as the
top
of
E
. Now fix a
point
P
= (
x, y
)
in
D
, and let
z
go vertically along
the red line from
P
up to the black dot at the top.
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 Spring '07
 TextbookAnswers
 Integrals, red line, dz dy dx

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