Graphics Programs Reference
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the weights are all symmetric about the coordinate axes. It followsthat the points
labeled a contribute equal amounts to I ; the same istruefor the points labeled b .
Therefore,
555 556) 2 cos 2 π
(0
.
774 597)
2
I
=
4(0
.
888 889) cos π
(0
.
774 597)
2
cos π
(0)
2
+
4(0
.
555 556)(0
.
888 889) 2 cos 2 π
(0)
2
+
(0
.
=
1
.
623 391
2
The exact value of the integral is 16
1
.
621 139.
EXAMPLE 6.15
y
4
3
4
3
1
2
x
1
1
4
= A
Utilize gaussQuad2 to evaluate I
f ( x
,
y ) dx dy over the quadrilateral shown,
where
2) 2 ( y
2) 2
f ( x
,
y )
=
( x
Use enough integrationpoints foran“exact”answer.
Solution The requiredintegration orderis determinedbythe integrand in Eq. (6.45):
1
1
I
=
f [ x (
ξ,η
)
,
y (
ξ,η
)] |
J (
ξ,η
)
|
d
ξ
d
η
(a)
1
1
We note that
|
J (
ξ,η
)
|
, definedinEqs. (6.44), is biquadratic.Since the specified f ( x
,
y )
is also biquadratic, the integrand in Eq. (a) is apolynomialofdegree 4 in both
ξ
and
η
3) issufficientforan“exact”result. Here is the
MATLAB command that performs the integration:
. Thusthird-orderintegration ( n
=
>>I=gaussQuad2(@fex6
_
15,[0;4;4;1],[0;1;4;3],3)
I=
11.3778
>>
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