is defined on a rectangular box:
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Just as was done with single integrals for functions of one variable and
double integrals for functions of two variables, we can define triple
integrals for functions of three variables. Lets first deal with the simplest
case where
is defined on a rectangular box:
The first step is to divide B into sub-boxes. we do this by dividing the
interval
![$[a,b]$](16.7 Triple Integrals__3.png)
into
subintervals
![$[x_{i-1},x_{i}]$](16.7 Triple Integrals__5.png)
of equal width
dividing
![$[c,d]$](16.7 Triple Integrals__7.png)
into
subintervals of equal width
and
dividing
![$[r,s]$](16.7 Triple Integrals__10.png)
into
subintervals of equal width
The planes though the endpoints of these subintervals parallel to the
coordinate planes divide the box B into lmn sub-boxes.
Hence each sub-box has the volume
This is demonstrated graphically below:
picture of Something like figure 1 on page 1043
Then we form the Triple Riemann sum:
where the sample point
is
in
Now from the definition of the integral we simply take the limit to find the
definition of the triple integral:
Just as for double integrals, the practical method for evaluating triple integrals is to express them as iterated integrals as in the following theorem:
If
is continuous on the rectangular box
then
The iterated integral on the right side of Fubini's Theorem means that we integrate first with respect to x (keeping y and z fixed), then we integrate with respect to y (keeping z fixed), and finally we integrate with respect to z. There are five other possible orders in which we can integrate, all of which five the same value. See the example below:
Now we define the triple integral over a general bounded region E in three
dimensional space. To do this we use the same procedure we did when defining
two dimensional integral over general regions. We restrict our attention to
continuous functions
and to certain simple types of regions. A solid region E is said to be of type
1 if it lies between the graphs of two continuous functions of x and y,
where D is the projection of E onto the xy-plane. An example of E is shown below (notice D):
Figure 2 on 1044 (or something like them) with labels.
A triple integral of this region is defined below:
In particular, if the projection D of E onto the xy-plane is a type I plane region, then:
This region is shown below (notice D):
Picture such as figure 3 on page 1044 labels
The triple integral then becomes:
If D is a type II plane region, then:
This type of region is shown below:
Picture such as figure 4 on page 1045:
The triple integral then becomes:
A solid region E is of type 2 if it is of the form:
In this type of region, D is the projection of E onto the yz-plane. This is shown below:
picture such as figure seven on 1045
Then triple integral of this region is defined as:
Finally a type 3 region is of the form
In type 3 regions, D is the projection of E onto the xz-plane. This is shown below:
Picture such as figure 8 on page 104
The triple integral of this region is defined as:
If
for all points in E, then the triple integral represents the volume of E:
Most of the applications of double integrals can be extended to triple
integrals. For instance, if the density function of a solid object that
occupies the region E is
in units of mass per unit volume, at any given point
then its mass is:
and its moments around the three coordinate planes are
The center of mass is located at the point
,
where:
If the density is constant, the center of mass of the solid is called the centroid of E. The moments of inertia about the three coordinate axes are:
Just as with double integrals, the total electric charge on a solid object
occupying a region E and having charge density
is:
If we have three continuous random variables X, Y, and Z, their joint density function is a function of three variables such that the probability that (X,Y,Z) lies in E is:
In particular,
The joint density function satisfies
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