Civil Engineering Reference
In-Depth Information
The exact solution may be found from the following integral:
1
10
10
x
∫
x
1
10
dx
=
|
=
4 342944819 0 434294482
.
−
.
=
3 9808650337
.
0
ln
0
3.3
SiMPSOn'S RuLE
More accurate integration can be achieved by Simpson's rules credited
to Simpson (1750). Consider a function
f
(
x
) graphed between
x=-
∆
x
and
x=
∆
x
as shown in Figure 3.3. An approximation of the area under the
curve between these two points would be to pass a parabola through the
points and zero (three points). The general second-degree parabola con-
necting the three points is as follows:
=
()
=++
2
yf x x xc
∆
x
∆
x
ax
3
bx
2
(
)
∫
2
A
=
ax xcdx
+
+
=
+
+
cx
32
−
∆
x
−
∆
x
2
3
=
()
+
()
3
Aax
∆
2
c
∆
x
(3.4)
The constants
a
,
b
, and
c
are found using the three points (-
∆
x
,
y
i
),
(0,
y
i+
1
),
and
(
∆
x
,
y
i+
2
) as follows:
y=f(x)
(x
i+1
, y
i+1
)
(x
i
, y
i
)
(x
i+2
, y
i+2
)
y=ax
2
+bx+c
x
∆x
∆
x
Figure 3.3.
Simpson's rule.
