10
Triple Integrals
If you’ve understood the preceding discussion, then the concept of a triple integral should
be a natural extension of the concept of (single) integrals and double integrals. We will now
be speaking of functions of three variables, and our domains of integration will be three
dimensional. As we’ve done for double integrals, we’ll avoid rigorous deFnitions, and instead
give outlines which will (we hope) enable you to construct the integrals you need.
If the domain of integration (call it
D
) is a rectangular box, with
x
∈
[
a
1
,a
2
]
,y
∈
[
b
1
,b
2
]
,z
∈
[
c
1
,c
2
]
,
then the jump from two variables to three is an easy one. We divide all
three axes into intervals of lengths
∆
x
,
∆
y
,and
∆
z
. In each of the resulting small boxes we
choose a point
(
x
∗
i
∗
j
∗
k
)
, evaluate the function (let’s call it
f
)atthatpo
in
t
,andmu
lt
ip
ly
the result by the volume of the box
∆
V
=
∆
x
∆
y
∆
z
(see ±igure 1). We then sum the results
±igure 1:
for every one of these boxes within the domain. The limit of the result as
∆
x,
∆
y,
∆
z
all
approach zero is our integral,
ˆ
D
f
(
x, y, z
)
dV
. Of course, there are now
six
possible orders
of integration, and for rectangular domains these can all be interchanged freely:
ˆ
D
f
(
x, y, z
)
dV
=
ˆ
c
2
c
1
ˆ
b
2
b
1
ˆ
a
2
a
1
f
(
x, y, z
)
dxdydz
=
ˆ
c
2
c
1
ˆ
a
2
a
1
ˆ
b
2
b
1
f
(
x, y, z
)
dydxdz
=
ˆ
b
2
b
1
ˆ
c
2
c
1
ˆ
a
2
a
1
f
(
x, y, z
)
dxdzdy
&c.
.
±or domains which are
not
rectangular, we can extend the concept of “Type I”, “Type II”, and
“Type III” regions.
.. but we can now distinguish between at least
seven
types! Numbering
them isn’t necessary; if we grasp the concept we should be able to write down an appropriate
integral.
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View Full DocumentExample 1:
Suppose the domain of integration can be described as
D
=
{
(
x, y, z
)

a
≤
x
≤
b,
φ
1
(
x
)
≤
y
≤
φ
2
(
x
)
,
γ
1
(
x, y
)
≤
z
≤
γ
2
(
x, y
)
}
, as shown in Figure 2 (this is the
threedimensional “Type I” region). This isn’t easy to show with a static 2D image, but what
we’re describing is a solid whose front and back sides are ±at (they are in the planes
x
=
a
and
x
=
b
), and whose left and right sides are curved in one dimension only, meaning that
you could easily mold a sheet of paper to them with no crumpling. The top and bottom,
meanwhile, may curve in two dimensions (they are unrestricted
surfaces
). The integral of a
Figure 2:
function
f
over such a domain can be expressed as
ˆ
D
f
(
x, y, z
)
dV
=
ˆ
b
a
ˆ
φ
2
(
x
)
φ
1
(
x
)
ˆ
γ
2
(
x,y
)
γ
1
(
x,y
)
f
(
x, y, z
)
dzdydx.
(1)
To understand why it must have this form, consider that a
deFnite integral
is a number, so
the last pair of limits of integration we use must be constants. That is, the outer integral must
be the integral in
x
, from
x
=
a
to
x
=
b
. Now, for each value of
x
in the interval
[
a, b
]
,the
values of
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 Winter '13
 Calculus, Integrals, Sin, Coordinate system, Spherical coordinate system, Polar coordinate system, xz dzdydx

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