Civil Engineering Reference
In-Depth Information
The exact solution may be found by the integral:
10
()
=
[
]
=
10
∫
log
xdxxxx e
log
−
log
5 65706 0 43429
.
+
.
=
6 09135
.
1
1
3.5
DOubLE intEgRAtiOn bY SiMPSOn'S
OnE-tHiRD RuLE
When using double integration by Simpson's one-third rule, weighting
is applied in both directions and is then multiplied by the spacing in both
directions. The following is a general weighting array for four strips:
14241
416816 4
28482
416816 4
14241
For two strips, the weighting array is the following:
141
4164
141
Any even set of strips will follow the same pattern and this could also
be done using any other type of integration. Using the trapezoidal rule
would be less accurate, but could do any number of strips and Simpson's
three-eighths rule would require a multiple of three strips in each direc-
tion. The summation of the weighting array multiplied by
f
(
x
,
y
) is used in
the following equation to obtain the volume.
hxhy
(
)
=
9
∑
∫
∫
=
fxyV
,
The terms
hx
and
hy
are the spacing in the x and y directions, respectively.
Example 3.6
Double integration by Simpson's rule
Determine the volume under the hyperbolic paraboloid from
x
= 0 to 8 and
y
= 0 to 8 for 0
=
16
z - xy
using Simpson's one-third rule with four strips
in each direction.
