Question

Apply Trapezoidal Rule, Simpsons 1/3 Rule and Trapezoidal Rule to compute

   a
I=∫   p/q+x^2
   0


Assume: i) the constant “a”- in between 1.1 to 1.5
        ii) the constant “p” – in between 2 to 5.
        iii) the constant “q” – in between 6 to 9.
         iv) the increment “h” – in between 0.025 to 0.1

Answer 1

p = 4;  q = 7;
f = @(x) p./(q+x.^2);

a = 1.5;
n = 20;  % number of intervals
h = a / n;

% Midpoint Rule:
I_m = 0;
for i=1:n
    x = h*(i-0.5);
    I_m = I_m + f(x);
end
I_m = I_m * h;

% Trapezoid Rule:
I_t = 0;
for i=1:n
   x1 = h*(i-1);
   x2 = h*i;
   I_t = I_t + (f(x1) + f(x2))/2;
end
I_t = I_t * h;

% Simpson's 1/3 Rule:
I_s = 0;
for i=1:2:n
    x1 = h*(i-1);
    x2 = h*i;
    x3 = h*(i+1);
    I_s = I_s + (f(x1) + 4*f(x2) + f(x3))/3;
end
I_s = I_s * h;

fprintf('Midpoint value:  %f\n', I_m);
fprintf('Trapezoid value: %f\n', I_t);
fprintf("Simpson's value: %f\n", I_s);