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TRAPEZOIDAL AND SIMPSONS RULE

General Objective : To determine the area of an uninformed plane using

Trapezoidal and Simpsons Rule.

Specific Objectives : On completion of this unit, you should be able to:-

find the area of a plane using the Trapezoidal rule.

find the area of a plane using the Simpsons rule

UNIT 4

OBJECTIVES

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4.0 INTRODUCTION

Using numerical analysis, function y can be determined for some values of x in an

experiment data or calculation of function. These data can be retrieved from experiments

or calculated from given function. These data are filled in the table form for function

y = f(x) then its is equivalent to the integral of function y = f(x) within x = a and x = b.

X X0 X1 X2 Xn

Y Y0 Y1 Y2 yn

Using x0=a and xn=b the values of x are constant h;

i.e

h=xi+1 xi i = 0,1, n 1

Figure 6.1 shows the values for yi is a function of xi.. Integrating the function of f

between x = a and x = b, you will find the area graph and the x-axis. Geometrically this

area can be estimated from the sum of area from rectangles width h and height yi

(i=0,1,.,n-1) shown in the figure. 6.2

y1 yn-2

y2 y3

yn-1

yn

y0

a=x0 x1 x2 x3 xn-2 xn-1 xn

Fig. 6.1

INPUT

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Mathematically, we write the equation as

Sum of area = hy0 + hy1 + hy2 + . + hyn-2 + hyn-1

or

Area = h

1

0

n

i

iy

In other words, we have to integrate as

b

a

dxxf )( h

1

0

n

i

iy

This is just a rough estimate and it is best to choose a small value of h

(distance between xn).

y3 y2

y1

Yn-2

Yn-1

y0

a=x0 x1 x2 x3 xn-2 xn-1 xn

Fig 6.2

yn

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4.1 TRAPEZOIDAL RULE

Using this method, the area under a function curve can be substituted with the

area within limits of adjacent yi. This limit forming a trapezium and its area can be

defined as in figure 4.3 below:.

The area for each trapezium with corresponding heights yi and yi+1 is given as

Area = iyh

(2

+ )1iy ; i = 0,1,2,.n-1

INPUT

y3

x0 x1 x2 x3 xn-2 xn-1 xn

y2 yn-2

yn-1

y0

yn y1

Figure 6.3

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The sum area of all trapezium between x0 and xn as

Sum area = )(2

)(2

...)(2

)(2

)(2

112322110 nnnn yyh

yyh

yyh

yyh

yyh

= )22......22(2

12210 nnn yyyyyyh

This sum of area can be summarized as

)22.....22(2

)( 12210 b

a

nnn yyyyyyh

dxxf

yi= f(x), x0=a, xn= b and h = xi+1 xi (i=0,1,2,.,n-1)

Example 4.1 :

Using the trapezoidal rule for y = 2x2 we can calculate the values of I

I= 10

0

)( dxxf

For values of )( ii xfy are calculated in the table below:

X 0 2 4 6 8

10

Y 0 8 32 72 128

200

Solutions :

Using the trapezoidal rule,

)22.....22(2

)( 12210 b

a

nnn yyyyyyh

dxxf

for h = 2, then :

I 200)12872328(202

2

= 680.0

This answer is just estimation. Using differentiation, the real value is 2000/3 = 666.7.

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ACTIVITY 4a

1. Given the differentiation of21

1

x is tan

-1 x. Comparing tan

-1 s with

s

x

dx

0

21

for s = 0.1,0.2,..,2.0. and h = 0.05 (Use Trapezium rule)

2. Calculate s

dxx

1

1 using trapezium rule for s = 2, s = 3, s= 3

s = 4, s = 5 and h = 0.025. Compare your answer with ln s.

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FEEDBACKS 4a

1 Integrate 1

1

2 x

S Trapezium Rule Tan-1

s

0.1 0.09963 0.09966

0.2 0.19732 0.19739

0.3 0.29135 0.29145

0.4 0.38038 0.38050

0.5 0.46351 0.46364

0.6 0.54028 0.54041

0.7 0.61059 0.61072

0.8 0.67462 0.67474

0.9 0.73270 0.73281

1.0 0.78529 0.78539

1.1 0.83289 0.83298

1.2 0.87597 0.87605

1.3 0.91503 0.91510

1.4 0.95048 0.95054

1.5 0.98273 0.78279

1.6 1.01214 1.01219

1.7 1.03903 1.03907

1.8 1.06366 1.06369

1.9 1.08628 1.08631

2.0 1.10712 1.17014

2 Integrate x

1

S Trapezium rule Ln S

2.0 0.69319 0.69314

3.0 1.09866 1.09861

4.0 1.338634 1.38629

5.0 1.60949 1.60943

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4.2 SIMPSONS RULE

In the following equation

.....6

)2)(1(

2

)1(0

3

0

2

00

fppp

fpp

fpfy

if you integrate between x0 and x2, you will find

2

0

2

0

x

x

ydphydx

2

0

0

3234

0

223

0

2

0 ...)6

1

6

1

24

1()

4

1

6

1(

2

1

fpppfppfppfh

........0

3

122 0

3

0

2

00 ffffh

and if the first 3 terms are taken, you will find

0

3

003

122

2

0

fffhydx

x

x

=

012010 2(

3

1)(22 ffffffh

= )4( 210 fffh

and in this formula, the first term involved 04 f is ignored as the coefficient of 0

3 f = 0

It can be shown that this term is 04

90f

h and you can expect Simpsons rule to be

more precise than the trapezium rule.

On implementing it, you have to group y in a group of threes. In between a darker

line is marked. On every three points, two marked lines will represent the area

.),........4(3

),4(3

),4(3

654432210 yyyh

yyyh

yyyh

INPUT

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The sum of area = ),.....4(3

)4(3

)4(3

654432210 fffh

fffh

fffh

= .....)42424(3

543210 ffffffh

This rule can be expanded for all values of x ; each has to consist 2 marked lines (for

every three points). If the number of marked lines is n, the rule will follow.

nx

x

nnnn ffffffffh

ydx

0

)424....424(3

1233210

Example 4.2 :

Integrate y= 2x2 using Simpsons rule if I =

10

0

)( dxxf

Solution :

with h = 1, you have 10 marked lines with 11 points.

Fig 6.4

x0 x1 x2 x3 x4 .

Y4

Y3

Y2

y1 y0

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X 0 1 2 3 4 5 6 7 8 9

10

Y

0

2

8

18

32

50

72

98

128

162

200

Using Simpson rule, you will have :

200)162(4)128(2)98(4)72(2)50(4)32(2)18(4)8(2)2(403

1I

3

2000I

7.666I

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ACTIVITY 4b

1. Given the differentiation of21

1

x is tan

-1 x. Comparing tan

-1 s with

s

x

dx

0

21 for s

= 0.1,0.2,..,2.0. and h = 0.05 [Use Simpson rule)

2. Calculate s

dxx

1

1 using Simpsons rule for s = 2, s = 3, s= 3

s = 4, s = 5 and h = 0.025. Compare your answer with ln s.

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FEEDBACK 4b

1. Integrate 1

1

2 x

S Simpson rule Tan-1

s

0.1 0.09967 0.09966

0.2 0.19740 0.19739

0.3 0.29146 0.29145

0.4 0.38051 0.38050

0.5 0.46365 0.46364

0.6 0.54042 0.54041

0.7 0.61073 0.61072

0.8 0.67474 0.67474

0.9 0.73282 0.73281

1.0 0.78540 0.78539

1.1 0.83298 0.83298

1.2 0.87606 0.87605

1.3 0.91510 0.91510

1.4 095055 0.95054

1.5 0.98279 0.78279

1.6 1.01220 1.01219

1.7 1.03907 1.03907

1.8 1.06370 1.06369

1.9 1.08632 1.08631

2.0 1.10715 1.10714

2. Differentiate x

1

S Simpson rule ln s

2.0 0.69315 0.69314

3.0 1.09861 1.09861

4.0 1.38629 1.38629

5.0 1.60944 1.60943

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1. Using Simpsons rule, integrate 6

2

.dxy when y=f(x)

2. Calculate dxex

10

0

2

. For x = 10 , the values of integration did not contribute

much , then the answer is best in the form of 2

. Use h = 1, h =0.5 and h = 0.1.

3. Estimate dxx12

0

2 using

a) Trapeziums rule for n = 12

b) Simpsons rule for n = 6

4. Estimate 3

0 31 x

dx using

a) Trapeziums rule for n = 6

b) Simpsons rule for n = 3

SELF ASSESSMENT

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FEEDBACK TO SELF ASSESSMENT

1. 44.3 unit2

2. Intgration for 2xe

h Trapezium rule Simpson rule

1.0 0.886318602 0.836214302

0.5 0.886226926 0.886196367

0.1 0.886226926 0.886226924

3 a) 578

b) 576

4. a) 1.15

b) 1.16