Closed NewtonCotes Integration
James Keesling
This document will discuss NewtonCotes Integration.
Other methods of numerical
integration will be discussed in other posts. The other methods will include the Trapezoidal
Rule, Romberg Integration, and Gaussian Integration.
1
Principle of NewtonCotes Integration
We only cover the NewtonCotes closed formulas.
The interval of integration [
a, b
] is
partitioned by the points.
a, a
+
b

a
n
, a
+ 2
·
b

a
n
, . . . , b
We estimate the integral of
f
(
x
) on this interval by using the Lagrange interpolating
polynomial through the following points.
(
a, f
(
a
))
,
a
+
b

a
n
, f
a
+
b

a
n
, . . . ,
(
b, f
(
b
))
The formula for the integral of this Lagrange polynomial simplifies to a linear combi
nation of the values of
f
(
x
) at the points
x
i
=
a
+
i
·
b

a
n
i
= 0
,
1
,
2
, . . . , n
.
In the next section we give a method for calculating the coefficients for this linear
combination.
2
The NewtonCotes Closed Formula
We wish to estimate the following integral.
Z
b
a
f
(
x
)
dx
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
We use the value of the function at the following points
{
a
+
i
·
b

a
n
i
= 0
,
1
,
2
, . . . , n
}
.
Our estimate will have the following form.
Z
b
a
f
(
x
)
dx
=
A
0
·
f
(
a
) +
A
1
·
f
a
+
b

a
n
+
A
2
·
f
a
+ 2
·
b

a
n
+
· · ·
+
A
n
·
f
(
b
)
So, what values should we use for the coefficients and how can we calculate them?
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 JURY
 Calculus, Numerical Analysis, normalized coeﬃcients

Click to edit the document details