Newton-Cotes numerical integration methods provide a systematic approach for approximating definite integrals using polynomial interpolation. This document explores various quadrature formulas derived from the Newton-Cotes framework, including the Trapezoidal Rule and Simpson's Rules. It is designed for students and professionals in mathematics and engineering, offering insights into numerical techniques for integration. Key topics include the derivation of formulas, application examples, and error analysis, making it a valuable resource for those studying numerical methods in calculus or preparing for related examinations.

Key Points

  • Explains Newton-Cotes numerical integration methods for approximating integrals.
  • Covers Trapezoidal Rule and Simpson's Rules with derivations and examples.
  • Includes error analysis for various numerical integration techniques.
  • Designed for students in mathematics and engineering fields.
Archita Gupta
16 pages
Biology
Archita Gupta
16 pages
Biology
124
/ 16
Chapter
5
Numerical Integration
Numerical integration is the process of evaluating a definite
integral from the values of the function given in tabular form
and when this process is applied to the integration of a function
of single variable, and then the process is called quadrature.
The numerical integration of the tabulated value is solved by
representing f(x) by an interpolation formula and then
integrating it between the given intervals.
A General quadrature formula:
Let
(
)
b
a
I
f x dx=
∫
Where f(x) is obtained by an interpolation formula in the
interval (a, b) which is divided into n sub-intervals of equal
width h. Let x
0
= a, x
1
=a + h, x
2
=a + 2h ............... x
n
= a+ nh =
b and corresponding values of f(x) be y
0
, y
1
, y
2
........ y
n
.
The number of ordinates is (n+1).
If y = f(x) = f(x
0
+ph) then
0
x x
p
h
−
=
dx
dp dx hdp
h
⇒ = ⇒ =
By Newton's forward interpolation formula
2 3
0 0 0 0
( 1) ( 1)(p 2)
...............
2
! 3!
p p p p
y y p y y y
− −
= + ∆ + ∆ + ∆ +
62 A Text Book of Numerical Technique
Integrating both sides.
0
0
2 3
0 0 0 0
0
4 5
0 0
6
0
( 1) ( 1)(p 2)
2! 3!
( 1)(p 2)(p 3) ( 1)(p 2)(p 3)(p 4)
+
4! 5!
( 1)(p 2)(p 3)(p 4)(p 5)
+ .........
6!
x nh
n
x
p p p p
ydx y p y y y
p p p p
y
y
p p
y hdp
+
− − −
= + ∆ + ∆ + ∆ +
− − − − − − −
∆
+ ∆ +
− − − − −
∆ +
∫ ∫
2 3 2
2 3
0 0 0 0
0
4 3 2 4 5 4 3 2 5
0 0
6 5 4 3 2 6
0
3 2
2! 3!
1 1
+( 6 11 6 ) ( 10 35 50 24 )
24 120
1
+( 15 85 225 274 120 ) ......... [
720
n
p p p p p
n y p y y y
p
p
p
p
y
p
p
p
p
p
y
p p p p p y dp x
− − +
= + ∆ + ∆ + ∆ +
−
+ − ∆ + − + − + ∆ +
− + − + − ∆ +
∫
0
0
]
=
On integrating with respect to p we get,
3
2 3 2 4
2 3 2
0
0 0 0
4
5 4 3
2
0
5
6 4 3
5 2
0
6
7 6 4 3
5 2
0
0
2 6 4 4 6
3 11
3
5 2 3 24
35 50
2 12
6 4 3 120
15 225 274
17 60 ............
7 6 4 3 720
n
y
p p p p
h py y y p p
y
p p p
p
y
p p p
p p
y
p p p p
p p
∆
= + ∆ + − ∆ + − +
∆
+ − + −
∆
+ − + − +
∆
+ − + − + − +
Numerical Integration 63
On Putting p = n, we get
3
2 3 2 4
2 3 2
0
0 0 0
4
5 4 3
2
0
5
6 4 3
5 2
0
6
7 6 4 3
5 2
0
2 6 4 4 6
3 11
3
5 4 3 24
35 50
2 12
6 4 3 120
15 225 274
17 60 ............
6 6 4 3 720
y
n n n n
n ny y y n n
y
n n n
n
y
n n n
n n
y
n n n n
n n
∆
= + ∆ + − ∆ + − +
∆
+ − + −
∆
+ − + − +
∆
+ − + − + − +
(1)
This is Newton-cotes quadrature formula and from this we
deduce different quadrature rules by taking n = 1, 2, 3,
6..................
Trapezoidal Rule:
We have to find the area of the region ABCD by trapezoidal
Rule. Base AB is divided into n equal parts. First of all we find
out the area of the Ist division and then area of the remaining
divisions.
On taking n = 1, in the equation (1) the figures will be a
trapezium and the curve will be a straight line passing through
(x
0
, y
0
) and (x
1
, y
1
) and other integrals will be zero.
Putting n = 1 in (1)
0
0
0 0 0 1 0
1 1
(
)
(
)
2 2
x h
x
f
x dx h
y
y
h
y
y
y
+
= + ∆ = + −
∫
[ ]
0 1
0 1
.........
(
1
)
2 2 2
y
y h
h y y
= + = +
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End of Document
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FAQs

What are the Newton-Cotes numerical integration methods?
Newton-Cotes numerical integration methods are techniques used to approximate the value of definite integrals through polynomial interpolation. These methods utilize equally spaced points to derive formulas that can estimate the area under a curve. The most common forms include the Trapezoidal Rule and Simpson's Rule, which differ in their approach to approximating the function being integrated. The document provides a detailed exploration of these methods, including their derivation and application.

How does the Trapezoidal Rule work in numerical integration?
The Trapezoidal Rule approximates the area under a curve by dividing it into trapezoids rather than rectangles. By calculating the area of each trapezoid formed between two points on the curve, the method sums these areas to estimate the integral. This approach is particularly useful for functions that are linear or nearly linear over small intervals. The document outlines the mathematical formulation of the Trapezoidal Rule and provides examples to illustrate its application.

What is Simpson's Rule and how is it applied?
Simpson's Rule is a numerical integration method that approximates the integral of a function by fitting parabolas to segments of the curve. It requires an even number of intervals and uses the values of the function at these points to create a quadratic polynomial that estimates the area under the curve. This method is more accurate than the Trapezoidal Rule for smooth functions, as it accounts for curvature. The document details the derivation of Simpson's Rule and includes practical examples for clarity.

What are the advantages of using numerical integration methods?
Numerical integration methods, such as those discussed in this document, provide practical solutions for evaluating integrals that are difficult or impossible to solve analytically. They allow for the approximation of areas under curves for a wide range of functions, making them essential in fields like engineering, physics, and finance. These methods also enable the handling of real-world data where exact solutions are not feasible. The document emphasizes the importance of understanding these techniques for students and professionals alike.

What is the significance of error analysis in numerical integration?
Error analysis in numerical integration is crucial for understanding the accuracy and reliability of the approximation methods used. It helps identify how close the numerical result is to the actual value of the integral and provides insights into the factors affecting this accuracy, such as the choice of method and the number of intervals used. The document discusses various error estimation techniques, allowing readers to assess the performance of different numerical integration methods.