Civil Engineering Reference
In-Depth Information
The exact solution may be found by the integral:
p
p
2
2
(
)
(
)
∫
2
2
∫
21
n
−
21
n
−
sincos
x
+
x dx
=
sin
x
+
cos
x dx
0
0
(()
…
()
()
…+
()
…
()
()
…+
62
35721
24
n
2462
35721
n
2
3
2
3
4
3
=
)
+
)
=+=
(
(
n
n
3.4
gAuSSiAn QuADRAtuRE
The main difference of Gaussian quadrature from the previous methods
is that the interval to be integrated is not divided into strips. Instead, a
central point is used to determine the best places to evaluate the function.
The Gauss points indicate how far from the central point to go and then
each point is weighted. The derivation of this method is not included
here, but can be found in many advanced mathematics textbooks. The
method is named for Carl Friedrich Gauss (1801). The following is a gen-
eral equation that shows the process for
n
Gauss points for integration:
b
n
=
()
=
(
)
∑
=1
∫
Af xdxsfs sx w
±
1
i
i
i
a
s
ba
−
=
2
ba
+
s
=
1
2
The number of points used should closely match the degree of the equation
to integrate. Table 3.11 shows some of the Gaussian quadrature points,
x
i
,
and their weights,
w
i
.
Example 3.5.
Gaussian quadrature
Determine the area under the curve from 1 to 10 for
y =
log
10
x
using
Gaussian quadrature with 2, 3, and 4 points.
b
n
=
()
=
(
)
∑
=1
∫
Af xdxsfs sx w
±
1
i
i
i
a
s
ba
−
10 1
2
−
=
=
=
45
.
2
ba
+
10 1
2
+
s
=
=
= 5.
1
2
