Civil Engineering Reference
In-Depth Information
Similarly, Simpson's three-eighths rule can be derived using three strips
and a third-degree parabola. The following is the solution:
=
()
=+++
yf x x x xd
3
2
3
∆
x
3
∆
x
2
ax
4
bx
3
cx
2
2
(
)
∫
3
2
A
=
ax x xddx
+ ++
=
+
+
+
dx
43
2
−∆
3
2
x
−
3
2
∆
x
9
4
=
()
+
()
3
Abx
∆
3
d
∆
x
(3.5)
The constants
a
,
b
,
c
,
and
d
are found using the four points (-3
∆
x/
2,
y
i
),
(-
∆
x/
2,
y
i+
1
), (
∆
x/
2,
y
i+
2
), and (3
∆
x/
2,
y
i+
3
) as follows:
3
2
ya
x
−
3
2
∆
−
3
2
∆
x
−
3
2
∆
x
+
+
+
=
b
c
d
i
3
2
y a
x
∆
−
∆
x
−
∆
x
−
+
=
+
b
+
c
d
i
+
1
2
2
2
3
2
∆
x
∆
x
∆
x
+
+
y
a
b
c
d
=
+
i
+
2
2
2
2
3
2
3
∆
x
3
∆
x
3
∆
x
y
=
a
+
b
+
c
+
d
i
+
3
2
2
2
Solve the four equations with four unknowns and then substitute these
back into Equation 3.5 to achieve the following:
3
∆
x
yy y
(
)
A
=
+
3
+
3
+
y
i
i
+
1
i
+
2
i
+
3
8
The general form with
n
strips is as follows:
3
∆
x
nn
−
1
,
n
−
2
∑∑
A
=
y
+
3
y yy
+
2
+
1
i
i
n
+
1
8
i
=
2356
,,,
i
=
47
,
For an odd number of strips, both the one-third and three-eighths rules
must be used. The three-eighths rule is used to obtain the area contained
in three strips under the curve and then the one-third rule is used for the
remaining
n
-3 strips.
