Civil Engineering Reference
In-Depth Information
Example 3.7
Double integration by Gaussian quadrature
Determine the volume under the hemisphere from
x
= −4 to 4 and
y
= −4
to 4 for 64
= x
2
+y
2
+z
2
using Gaussian quadrature with three points in each
direction.
Solving for
z
and the points:
2
2
z
=−−
64
x
y
=
−
()
=
44
2
s
ba
−
x
x
=
4
x
2
=
+
()
=
ba
+
44
2
s
=
x
x
0
x
1
2
=
−
()
=
s
ba
−
44
2
y
y
=
4
y
2
=
+
()
=
ba
+
44
2
y
y
s
=
0
y
1
2
Table 3.16 shows the set up and summation as follows with
x
i
,
y
i
given
values and
w
x
,
w
y
corresponding weights when using three points:
Table 3.16.
Example 3.7 Double integration by Gaussian quadrature
x =
s
x1
+ s
x
x
i
y =
s
y1
+ s
v
y
i
x
i
y
i
w
x
w
y
f(x,y)
w
x
*w
y
*f(x,y)
-0.774597
-0.774597 -3.098387 -3.098387 0.555556 0.555556
6.69328
2.065827
-0.774597
0
-3.098387
0
0.555556 0.888889 7.375636
3.642289
-0.774597
0.774597
-3.098387
3.098387
0.555556 0.555556
6.69328
2.065827
0
-0.774597
0
-3.098387 0.888889 0.555556 7.375636
3.642289
0
0
0
0
0.888889 0.888889
8
6.320988
0
0.774597
0
3.098387
0.888889 0.555556 7.375636
3.642289
0.774597
-0.774597
3.098387
-3.098387 0.555556 0.555556
6.69328
2.065827
0.774597
0
3.098387
0
0.555556 0.888889 7.375636
3.642289
0.774597
0.774597
3.098387
3.098387
0.555556 0.555556
6.69328
2.065827
29.153453
S
∑
4429 153
=
()
Vs
xy
=
.
=
466 45
.
