Exp-07

Course No: EEE 212

Experiment No: 07

Name of the experiment:

NUMERICAL INTEGRATION

Name: Md. Arifur Rahman

Date of performance: 29-04-2015 Student ID: 1206130

Date of submission: 06-04-2015 Department: EEE

Level 2 / Term 2

Section: B ( B-2 )

Partners ID: 1206121

Group No:15

EXERCISE 1 -

Integrate the function tabulated in Table over the interval

from x=1.6 to x=3.8 using composite trapezoidal rule with (a) h=0.2,

(b) h=0.4 and (c) h=0.6

X f(X) X f(X)

1.6 4.953 2.8 16.445

1.8 6.05 3.0 20.086

2.0 7.389 3.2 24.533

2.2 9.025 3.4 29.964

2.4 11.023 3.6 36.598

2.6 13.468 3.8 44.701

The data in Table 7.1 are for f (x) = ex . Find the true value of the integral and compare this with those found in (a), (b) and (c).

Solution clear all; clc;

close all;

x = [1.6 1.8 2.0 2.2 2.4 2.6

2.8 3.0 3.2 3.4 3.6 3.8];

y = [4.953 6.05 7.389 9.025

11.023 13.468 16.445 20.086

24.533 29.964 36.598 44.701];

n = length(x)-1;

actual = 39.748152;

sum = (y(1) + y(n+1))/2;

for j = 2 : n

sum = sum + y(j);

end

h = 0.2;

integral = h * sum;

error = abs(actual - integral);

fprintf('Actual Result =

39.748152\n');

fprintf('\nFor h = 0.2

calculated result = %f',

integral);

fprintf('\nFor h = 0.2 error

calculated = %f', error);

h = 0.4;

integral = h * sum;




integral);



h = 0.6;

integral = h * sum;




integral);



EXERCISE 2 (a) Integrate the function tabulated in Table 7.1 over the interval

from x=1.6 to x=3.6 using Simpsons composite 1/3 rule. (b) Integrate the function tabulated in Table 7.1 over the interval

from x=1.6 to x=3.4 using Simpsons composite 3/8 rule.

Solution (a)

clear all;

clc;

close all;

x = [1.6 1.8 2.0 2.2 2.4 2.6

2.8 3.0 3.2 3.4 3.6];

y = [4.953 6.05 7.389 9.025

11.023 13.468 16.445 20.086

24.533 29.964 36.598];

n = length(x)-1;

h = ( x(n+1) - x(1) )/n;

% Simpson 1/3 method

integral = 0;

for i = 1:2:n-1

integral = integral +

(y(i)+4*y(i+1)+y(i+2));

end

integral = (h/3)*integral

(b)

clear all;

clc;

close all;

x = [1.6 1.8 2.0 2.2 2.4 2.6

2.8 3.0 3.2 3.4];

y = [4.953 6.05 7.389 9.025

11.023 13.468 16.445 20.086

24.533 29.964];

n = length(x)-1;

h = ( x(n+1) - x(1) )/n;

%Simpson 3/8 method

sum = 0;

for i = 1:3:n-2

sum = sum + y(i) +

3*y(i+1) + 3*y(i+2) +

y(i+3);

end

integral = ((3*h)/8) * sum

EXERCISE 3

Solution (a)(i)

clear all;

clc;

close all;

a = -1;

b = 1;

n = 12;

h = (b-a)/n;

x = a;

f = inline('(1+X^2)^-1');

sum = f(x);

for i = 2:n

x = x + h;

sum = sum + 2*f(x);

end

x = b;

sum = sum + f(x);

integral = (h/2) * sum

(a)(ii)

clear all;

clc;

close all;

a = 0;

b = 4;

n = 12;

h = (b-a)/n;

x = a;

f = inline('(X^2)*exp(-X)');

sum = f(x);

for i = 2:n

x = x + h;

sum = sum + 2*f(x);

end

x = b;

sum = sum + f(x);


(b) Simpsons composite 1/3 rule

clear all;

clc;

close all;

% 3(b)

a = -1;

b = 1;

n = 12;

h = (b-a)/n;

x = a;

f = inline('(1+X^2)^-1');

sum = 0;

for i = 1:2:n-1

sum = sum + f(x+(i-1)*h) +

4*f(x+i*h) + f(x+(i+1)*h);

end


(b) Simpsons composite 3/8 rule

clear all;

clc;

close all;

a = 0;

b = 4;

n = 12;

h = (b-a)/n;

x = a;

f = inline('(X^2)*exp(-X)');

sum = 0;

for i = 1:3:n-2

sum = sum + f(x+(i-1)*h) +

3*f(x+i*h) + 3*f(x+(i+1)*h) +

f(x+(i+2)*h);

end

integral = (3*h/8) * sum

EXERCISE 4

Solution clear all;

clc;

close all;

a = 0;

b = 2;

toler = 0.249916*1e-3;

h = b-a;

f = inline('X*exp(-2*X^2)');

m1 = (f(b) + f(a))/2;

it1 = h*m1;

it0 = 0;

i = 1;

while abs(it1-it0) > toler

x = a + h/2;

for j = 1:i

m1 = m1 + f(x);

x = x + h;

end

i = 2*i;

h = h/2;

it0 = it1;

it1 = h * m1;

end

integral = it1

EXERCISE 5

Solution clear all;

close all;

clc;

format long;

actual = pi/4;

toler = 0.0001;

a = 0;

b = pi/2;

h = b-a;

sum = ((sin(16*x1))^2)/2;

it1 = h*sum;

diff = 1;

i = 1;

while diff > toler

x = a + h/2;

for j=1:i

sum = sum +

((sin(16*x))^2);

x = x+h;

end

i = i*2;

h = h/2;

it0 = it1;

it1 = h*sum;

diff = abs(it1-it0);

end

Cal_value = it1

deviation = ((abs(actual-

it1))/actual)*100

Exp-07

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