Civil Engineering Reference
In-Depth Information
Example 3.2
Romberg integration
Determine the third-order extrapolation for the area under the curve from
0 to 1 for
y=
10
x
using Romberg integration along with the trapezoidal rule.
n
∆
2
x
∑
A
=
y
+
2
y
+
y
1
i
n
+
1
i
=
2
The trapezoidal integration for one strip is as follows in Table 3.3:
Table 3.3.
Example 3.2 Romberg integration
x
0
1.000
y
1.0000
10.0000
1
2
1105.
(
)
=
A
=+
Two strips are shown in Table 3.4:
Table 3.4.
Example 3.2 Romberg integration
x
0
0.500
1.000
y
1.0000
3.1623
10.0000
1
4
123 1623
(
)
=
A
=+
(
)
+
.
10
4 33114
.
Four strips are shown in Table 3.5:
Table 3.5.
Example 3.2 Romberg integration
x
0
0.250
0.500
0.750
1.000
y
1.0000
1.7783
3.1623
5.6234
10.0000
1
8
121 7783 231623 256234
(
)
=
A
=+
(
)
+
(
)
+
(
)
+
.
.
.
10
4 01599
.
Eight strips are shown in Table 3.6:
Table 3.6.
Example 3.2 Romberg integration
x
0
0.125
0.250
0.375
0.500
0.625
0.750
0.875
1.000
y
1.0000
1.3335
1.7783
2.3714
3.1623
4.2170 5.6234 7.4989
10.0000
