Civil Engineering Reference
In-Depth Information
Example 3.3.
Simpson's one-third rule
Determine the area under the curve from
-p/
2
to p/
2 for
y = x
2
cos
(
x
) using
Simpson's one-third rule with four and eight strips.
n
n
−
1
∆
3
x
∑∑
A
=
y
+
4
y
+
2
yy
+
1
i
i
n
+
1
i
=
246
,,
i
=
3 57
,,
Four strips are shown in Table 3.7:
Table 3.7.
Example 3.3 Simpson's one-third rule
-
p
/2
-
p
/4
0
p
/4
p
/2
x
-1.57080
-0.78540
0.00000
0.78540
1.57080
y
0.00000
0.43618
0.00000
0.43618
0.00000
p
p
(
)
=
+
(
)
+
()
+
(
)
+
(
)
A
=
12
040 43618 20 4043618
.
.
0
3 48944
.
12
=
0 91353
.
3
Eight strips are shown in Table 3.8:
Table 3.8.
Example 3.3 Simpson's one-third rule
-
p
/2
-3
p
/8
-
p
/4
-
p
/8
p
/8
p
/4
3
p
/8
p
/2
0
x
-1.57080
-1.17810
-0.78540
-0.39270 0.00000 0.39270 0.78540 1.17810
1.57080
y
0.00000
0.53113
0.43618
0.14247
0.00000 0.14247 0.43618 0.53113
0.00000
p
24
040 53113 2043618 4014247 20 4014247
(
+
(
)
+
(
)
+
(
)
+
()
+
(
)
A
=
.
.
.
.
p
(
)
+
(
)
+=
(
)
=
+
2
0 43618 4053113
.
.
0
)
7 13352
.
0 933776
.
24
Perform Romberg extrapolation with these two integrations to get a more
exact solution as follows:
=
(
)
n
0 933776 16
.
1
−
−
0 913533
.
I
16
16
−
−
I
I
≅
h
2
h
1
=
0 935126
.
n
1
16
1
1
