Civil Engineering Reference
In-Depth Information
Four strips are shown in Table 3.2:
Table 3.2.
Example 3.1 Trapezoidal rule
x
p
0
p/4
p/2
3p/4
x
0
0.7854
1.5708
2.3562
3.1416
y=sin(x)
0
0.7071
1
0.7071
1.23E-16
p
p
(
)
=
A
=+
(
)
+
()
+
(
)
+
(
)
=
8
020 7071 21 207071
.
.
0
4 8284
.
1 8961
.
8
The exact solution may be found by the integral:
p
∫
()
=−
()
=−
()
−
()
sin
xdx
cos
x
|
p
cos
p
cos
0112
=+=
0
0
3.2
ROMbERg intEgRAtiOn
A more accurate integral can be obtained using Romberg's method
(Romberg 1955). If a function can be defined as a continuous mathe-
matical expression having continuous derivative
f′
(
x
) and
f″
(
x
), the
error of the trapezoidal rule is shown in Figure 3.2 and can be found
as follows:
Expanding
y
i+
1
in a Taylor series about
x
i
and letting
∆
x=h
as
follows:
2
3
y yh
yh yh
Higherorder termsi
′′
′′′
y
+
=+
′
+
i
+
i
+
i
1
i
i
2
!
3
!
The change in
y
between points
i
and
i+
1 is equal to the area under the
y′
curve between those two points, therefore the exact area in the strip is
as follows:
2
3
yh yh
Higherorder termsi
′′
′′′
y
−=
′
+
y
yh
i
+
i
+
i
+
1
i
i
2
!
3
!
fh fh
Higherorder termsi
′
2
′′
3
y
−=
y
ffh
+
i
+
i
+
(3.1)
i
+
1
i
i
2
!
3
!
