Question

[Computer Exercises 5.3.1] Find approximate values for the two integrals 1 dr 0 0 Use composite Simpson's rule with an error-Ί 10-5

please include matlab code!!

Answer 1

Code: Function1.m

clc
clear
a = 0; b = 1;
f = @(x) 4.* 1./(1 + x.^2);
n = 2; % you can also increase n
err = 1;
while err > 0.5*10^-5
h = (b - a) / (n - 1);
x = a:h:b;
wS = h/3 * [1 2*ones(1,n-2) 1] ;
wS(2:2:end) = 4*h/3 ;
IS1 = sum(f(x) .* wS);
I = quad(f, 0, 1);
err = abs(IS1 - I);
n = n + 1;
end

disp("Absolute error of f1 = "+num2str(err));
disp("Integral of f1 = "+num2str(IS1));

Output:

Absolute error of fl = 9.0199e-07 Integral of f1 = 3.1416 x >

Code: Function2.m

clc
clear
a = 0; b = 1/sqrt(2);
f = @(x) 8 * (sqrt(1 - x.^2) - x);
n = 2; % you can also increase n
err = 1;
while err > 0.5*10^-5
h = (b - a) / (n - 1);
x = a:h:b;
wS = h/3 * [1 2*ones(1,n-2) 1] ;
wS(2:2:end) = 4*h/3 ;
IS2 = sum(f(x) .* wS);
I = quad(f, 0, b);
err = abs(IS2 - I);
n = n + 1;
end

disp("Absolute error of f2 = "+num2str(err));
disp("Integral of f2 = "+num2str(IS2));

Output:

Command Window Absolute error of f2 3.3594e-06 Integral of f2-3.1416 fx >>