Problem 7.42
Given:
By use of each numerical formula in Table 7.41, obtain four approximations for the integral of the
function
φ
=(1+rs)
1
over a triangle.
Let A=1.
Solution:
I
A
1
1
r s
⋅
+
⌠
⎮
⎮
⌡
d
=
A
1
=
J
2 A
⋅
=
J
i
1
2
J
⋅
=
1
=
For a 1point rule:
r
1
1
3
:=
s
1
1
3
:=
W
1
1
:=
I1
1
1
r
1
s
1
⋅
+
1
⋅
1
⋅
0.9
=
:=
For the first 3point rule:
n
3
:=
r
1
2
3
:=
s
1
1
6
:=
W
1
1
3
:=
J
1
1
:=
r
2
1
6
:=
s
2
1
6
:=
W
2
1
3
:=
J
2
1
:=
r
3
1
6
:=
s
3
2
3
:=
W
3
1
3
:=
J
3
1
:=
I3a
1
3
i
1
1
r
i
s
i
⋅
+
J
i
⋅
W
i
⋅
⎛
⎜
⎝
⎞
⎟
⎠
∑
=
0.924
=
:=
For the second 3point rule:
r
1
1
2
:=
s
1
0
:=
W
1
1
3
:=
J
1
1
:=
r
2
0
:=
s
2
1
2
:=
W
2
1
3
:=
J
2
1
:=
r
3
1
2
:=
s
3
1
2
:=
W
3
1
3
:=
J
3
1
:=
I3b
1
3
i
1
1
r
i
s
i
⋅
+
J
i
⋅
W
i
⋅
⎛
⎜
⎝
⎞
⎟
⎠
∑
=
0.933
=
:=
For the 4point rule:
r
1
1
3
:=
s
1
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 Fall '10
 Folkman
 Numerical Analysis, 3point rule

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