rectangular rule in matlab

main.m
function I = rectangular_rule(f,a,b,n)
% f is the function to integrate
% a and b are the limits of integration
% n is the number of rectangles to use
% I is the approximation of the integral

% Compute the width of each rectangle
dx = (b-a)/n;

% Initialize the sum of rectangle areas
sum_areas = 0;

% Compute the area of each rectangle and add it to the sum
for i=1:n
    x_i = a + (i-1/2)*dx; % x_i is the midpoint of the i-th rectangle
    f_i = feval(f,x_i); % Evaluate the function at x_i
    area_i = f_i*dx; % Compute the area of the i-th rectangle
    sum_areas = sum_areas + area_i; % Add the area to the sum
end

% Return the sum of the rectangle areas as the approximation of the integral
I = sum_areas;

% Example usage:
% Estimate the integral of f(x) = x^2 + 1 from x=0 to x=2 using 4 rectangles
% I = rectangular_rule(@(x) x.^2+1,0,2,4);
% The result should be I = 5.125
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