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The integration error is calculated using the error formula of Table 6.7 for
−
1
<
r
<
1. We
find for
r
=
1,
e
1
.
6
ξ
+
2
ξ
=
1
=
1
4
1
135
F
(
4
)
(
1
135
∂
6
4
.
r
)
=
0
.
02
(6)
∂ξ
4
=−
and for
r
1,
e
1
.
6
ξ
+
2
ξ
=−
1
=
1
4
1
135
F
(
4
)
(
1
135
∂
6
4
.
r
)
=
0
.
004
(7)
∂ξ
4
so that
0
.
004
<
R
n
<
0
.
02
(8)
Thus,
2
.
8
e
x
2
6
.
549
<
dx
<
6
.
573
(9)
1
.
2
Rectangular and Prism Regions
Integration of a function
F
(ξ
,
η)
over a rectangular region
−
1
≤
ξ
≤
1
,
−
1
≤
η
≤
1 can be
ξ
η
accomplished by choosing
m
and
h
integration points in the
and
directions, as in the case
=
=
η
of Fig. 6.38 for
h
m
3, evaluating the integral by holding
constant and integrating
over
ξ
,
and then holding
ξ
constant and integrating over
η
. This leads to
+
1
+
1
h
m
W
(
m
)
i
W
(
h
)
j
F
(ξ
,
η)
d
ξ
d
η
=
F
(ξ
i
,
η
)
(6.125)
j
−
1
−
1
j
=
1
i
=
1
Table 6.7 now applies in two directions for the determination of
W
(
h
)
j
and
W
(
m
i
according
to Gaussian quadrature. In this case, the total number of integration points in the domain
is
h
n
.
Similarly, for a right prism region, e.g., a brick configuration,
+
1
×
m
=
+
1
+
1
m
h
W
(
h
)
i
W
(
m
)
j
W
()
k
F
(ξ
,
η
,
ζ)
d
ξ
d
η
d
ζ
=
F
(ξ
i
,
η
j
,
ζ
k
)
(6.126)
−
1
−
1
−
1
k
=
1
j
=
1
i
=
1
where the total number of integration points is
×
h
×
m
=
n
.
FIGURE 6.38
Integration points for a square region with
h
=
m
=
3.
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