 # Triple integrals

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Triple integrals
```MULTIPLE INTEGRALS
y
1
dy
x+y =1
R
0
dx
0
1
x
Figure 6.2 The triangular region whose sides are the axes x = 0, y = 0 and
the line x + y = 1.
and that the order of integration is from right to left. So, in this example, the
integrand f(x, y) is ﬁrst to be integrated with respect to y and then with respect
to x. With the double integral expressed in this way, we will no longer write the
independent variables explicitly in the limits of integration, since the diﬀerential
of the variable with respect to which we are integrating is always adjacent to the
relevant integral sign.
Using the order of integration in (6.3), we could also write the double integral as
d x2 (y)
dy
dx f(x, y).
I=
c
x1 (y)
Occasionally, however, interchange of the order of integration in a double integral
is not permissible, as it yields a diﬀerent result. For example, diﬃculties might
arise if the region R were unbounded with some of the limits inﬁnite, though in
many cases involving inﬁnite limits the same result is obtained whichever order
of integration is used. Diﬃculties can also occur if the integrand f(x, y) has any
discontinuities in the region R or on its boundary C.
6.2 Triple integrals
The above discussion for double integrals can easily be extended to triple integrals.
Consider the function f(x, y, z) deﬁned in a closed three-dimensional region R.
Proceeding as we did for double integrals, let us divide the region R into N
subregions ∆Rp of volume ∆Vp , p = 1, 2, . . . , N, and let (xp , yp , zp ) be any point in
the subregion ∆Rp . Now we form the sum
S=
N
f(xp , yp , zp )∆Vp ,
p=1
190
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