in Simpsons rule have 5 ordinates than (n) equals to ?
Answers
Step-by-step explanation:
The Simpson's Rule formula with n=4 segments is written in the form. S4=Δx3[f(x0)+4f(x1)+2f(x2)+4f(x3)+(x4)]. The width of the subinterval is. Δx=b−an=π−04=π4
Answer:
Simpson’s Rule is a numerical method that approximates the value of a definite integral by using quadratic functions.
This method is named after the English mathematician Thomas Simpson
(
1710
−
1761
)
.
Simpson’s Rule is based on the fact that given three points, we can find the equation of a quadratic through those points.
To obtain an approximation of the definite integral
b
∫
a
f
(
x
)
d
x
using Simpson’s Rule, we partition the interval
[
a
,
b
]
into an even number
n
of subintervals, each of width
Δ
x
=
b
−
a
n
.
On each pair of consecutive subintervals
[
x
i
−
1
,
x
i
]
,
[
x
i
,
x
i
+
1
]
,
we consider a quadratic function
y
=
a
x
2
+
b
x
+
c
such that it passes through the points
(
x
i
−
1
,
f
(
x
i
−
1
)
)
,
(
x
i
,
f
(
x
i
)
)
,
(
x
i
+
1
,
f
(
x
i
+
1
)
)
.
Simpson's method of integration
Figure 1.
If the function
f
(
x
)
is continuous on
[
a
,
b
]
,
then
b
∫
a
f
(
x
)
d
x
≈
Δ
x
3
[
f
(
x
0
)
+
4
f
(
x
1
)
+
2
f
(
x
2
)
+
4
f
(
x
3
)
+
2
f
(
x
4
)
+
⋯
+
4
f
(
x
n
−
1
)
+
f
(
x
n
)
]
.
The coefficients in Simpson’s Rule have the following pattern:
1
,
4
,
2
,
4
,
2
,
…
,
4
,
2
,
4
,
1