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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
,
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
,
η)
d
ξ
d
η =
F
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
,
η
,
ζ)
d
ξ
d
η
d
ζ =
F
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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