Civil Engineering Reference
In-Depth Information
The exact solution may be found by the integral:
p
2
p
(
)
()
()
=
()
+−
∫
2
2
2
x
cos
x dx
2
xcos
xx x
2
sin
−
p
2
−
p
2
(
)
−−
(
)
=
=+
00467401
.
1003 004674011003
.
0 9348022005
.
Example 3.4.
Simpson's one-third and three-eighths rules
Determine the area under the curve from 0 to
p/
2 for
y = sin
3
x+cos
3
x
using
Simpson's three-eighths and one-third rules (in that order) with five strips.
Simpson's three-eighths rule is set up in Table 3.9:
3
∆
x
yy y
(
)
A
=
+
3
+
3
+
y
i
i
+
1
i
+
2
i
+
3
8
Table 3.9.
Example 3.4 Simpson's one-third and three-eighths rules
0
p
/10
p
/5
3
p
/10
2
p
/5
p
/2
x
0.00000
0.31416
0.62832
0.94248
1.25664
1.57080
y
1.00000
0.88975
0.73258
0.73258
0.88975
1.00000
3
80
130 88975 3073258
p
3
80
p
(
)
=
+
(
)
+
(
)
+
(
)
=
A
=
.
.
0 73258
.
6 59957 077
.
.7494
Simpson's one-third rule is set up in Table 3.10:
∆
3
x
(
)
A
=
y
+
4
y
+
y
i
i
+
1
i
+
2
Table 3.10.
Example 3.4 Simpson's one-third and three-eighths rules
0
p
/10
p/5
3p/10
2
p
/5
p
/2
0.00000
0.31416
0.62832
0.94248
1.25664
1.57080
1.00000
0.88975
0.73258
0.73258
0.88975
1.00000
p
p
(
)
=
+
(
)
+
(
)
=
A
=
0 73258 4088975
.
.
1
5 29158
.
0 554133
.
30
30
Adding the two together for a total area:
A
=
0 777494 0 554133 1 331627
.
+
.
=
.
