Newton Cotes Integration Method

Prepared By:- Pabari Shashikant.

INTRODUCTION

-In numerical analysis, the Newton–Cotes formulae, also called the Newton–Cotes quadrature rules or simply Newton–Cotes rules, are a group of formulae for numerical integration(also called quadrature) based on evaluating the integrand at equally spaced points.

-They are named after Isaac Newton and Roger Cotes.

-Newton–Cotes formulae can be useful if the value of the integrand at equally spaced points is given.

DESCRIPTION

It is assumed that the value of a function ƒ defined on [a, b] is known at equally spaced points xi, for i = 0, …, n, where x0 = a and xn = b.

Solved Using Newton-Cotes Formulae

There are two types of Newton–Cotes formulae,

1)The "closed" type.

2)The "open" type.

the closed newton-cotes formulae

In this type which uses the function value at all ʆ points.

The closed Newton–Cotes formula of degree n is stated as…

b n

∫ f(x) dx = ∑ wi.f(xi)

a i=0

where xi = h i + x0, with h (called the step size) equal to (xn − x0) / n = (b − a) / n.

The wi are called weights.

Closed Newton–Cotes Formulae

Degree Common name Formula Error term

1 Trapezoid rule

2 Simpson's rule

3 Simpson's 3/8 rule

4 Boole's rule

Trapezoid rules Trapezoidal Rule is based on the Newton-Cotes Formula that states if one

can approximate the integrand as an nth order polynomial

Then the integral of that function is approximated by the integral of that nth order polynomial.

Trapezoidal Rule assumes n=1, that is, the area under the linear polynomial,

-The trapezoidal rule works by approximating the region under the graph of the function as a trapezoid and calculating its area. It follows that ….

The function f(x) (in blue) is approximated by a linear function (in red).

Error analysis

The error of the composite trapezoidal rule is the difference between the value of the integral and the numerical result.

here exists a number ξ between a and b, such that

Method Derived From Geometry

The area under the curve is a trapezoid. The integral

trapezoidofAreadxxfb

a

)(

)height)(sidesparallelofSum(2

1

)ab()a(f)b(f 21

2)b(f)a(f

)ab(

Figure 2: Geometric Representation

f(x)

a b

b

a

dx)x(f1

y

x

f1(x)

Example 1

The vertical distance covered by a rocket from t=8 to t=30 seconds is given by:

a) Use single segment Trapezoidal rule to find the distance covered.

b) Find the true error, for part (a).c) Find the absolute relative true error, for part (a).

tEa

30

8

8.92100140000

140000ln2000 dtt

tx

Solution

2

)()()(

bfafabI

a)

8a 30b

tt

tf 8.92100140000

140000ln2000)(

)8(8.9)8(2100140000

140000ln2000)8(

f

)30(8.9)30(2100140000

140000ln2000)30(

f

sm /27.177

sm /67.901

Solution (cont)

2

67.90127.177)830(I

m11868

a)

b) The exact value of the above integral is

30

8

8.92100140000

140000ln2000 dtt

tx m11061

Solution (cont)

b) ValueeApproximatValueTrueEt

1186811061

m807

c) The absolute relative true error, , would be t

10011061

1186811061

t %2959.7

Open Newton–Cotes Formulas

Common name

step sizeFormula Error term

Degree

Rectangle rule, or

midpoint rule2

Trapezoid method

3

Milne's rule 4

No Name 5

The Open Newton-Cotes formula

In this type which not use the function values at the endpoints.

b n-1

∫ f(x) dx = ∑ wi.f(xi)

a i=1

The weights are found in a manner similar to the closed formula.

rectangular rules

The rectangle method also called the midpoint or mid-ordinate rule.

It computes an approximation to a definite integral, made by finding the area of a collection of rectangles whose heights are determined by the values of the function.

Formula:- (b - a) f1

Specifically, the interval over which the function is to be integrated is divided into equal sub intervals of length.

The rectangles are then drawn so that either their left or right corners, or the middle of their top line lies on the graph of the function, with bases running along the –axis,,,

Giving formula are,,,,,

where h=(b - a) / N And Xn=a + nh The formula for above gives for the Top-left corner approximation.

Animation Based Rectangular Method Graph

Midpoint Approximation Graph

Error Analysis

For a function which is twice differentiable, the approximation error in each section of the midpoint rule decays as the cube of the width of the rectangle.

for some in . Summing this, the approximation error for intervals with width is less than or equal to

N=1,2,3 where n + 1 is the number of nodes

n terms of the total interval,we know that so we can rewrite the expression:

for some in (a,b).

Application of Simpson’s Rules

Space Find the distance of the travel through velocity and

interval of time Find the Volume of the solid Calculate amount of earth that must be moved to fill a

depression or make a dam.

The Velocity v(km/min) of a moped which starts from rest is given at fixed intervals of time t(min) as follows:

km309.3372)2*804*(03220

0tsdistancerequiredtheHence

725202018v8v6v4v2E

80211322510v9v7v5v3v1O

000v10v1X

etc.25v3,18v2,10v1,0v0,2hHere

20

0rulesSimpson'by,2.E4.OX

3h

dtv200ts

vdtds

02511203229251810:v

2018161412108420:t

Estimate Approximately the distance covered in 20 minutes.

Sol. If s(km) be the distance covered in t(min), then

REAL TIME EXAMPLE OF SIMPSON’S METHOD

Thank You