how to calculate centroid of an area of a line and parabola in matlab

main.m
m = 2;          % Slope of the line
b = 5;          % y-intercept of the line
a = -1/4;       % Coefficient of x^2 in parabola equation
b_parabola = 3; % Coefficient of x in parabola equation
c = 2;          % Constant term in parabola equation

% Define the line and parabola as anonymous functions
f_line = @(x) m*x + b;
f_parabola = @(x) a*x.^2 + b_parabola*x + c;
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main.m
% Find the x-coordinates of the intersection points
a_quad = a;        % Coefficient of x^2 term
b_quad = b_parabola - m; % Coefficient of x term
c_quad = c - b;    % Constant term
x_intersect = roots([a_quad b_quad c_quad]);
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main.m
% Calculate the areas of the two parts of the bounded region
area_above = quadgk(@(x) f_parabola(x) - f_line(x), x_intersect(1), x_intersect(2));
area_below = quadgk(@(x) f_line(x) - 0, x_intersect(1), x_intersect(2));
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main.m
% Calculate the x-coordinate of the centroid
x_dist = @(x) abs(x - 0);  % The distance from the y-axis to the line is always x
x_c_above = (1/area_above) * quadgk(@(x) x_dist(x) * (f_parabola(x) - f_line(x)), x_intersect(1), x_intersect(2));
x_c_below = (1/area_below) * quadgk(@(x) x_dist(x) * f_line(x), x_intersect(1), x_intersect(2));
x_c = (area_above * x_c_above + area_below * x_c_below) / (area_above + area_below);
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main.m
% Calculate the y-coordinate of the centroid
y_c_above = (1/area_above) * quadgk(@(x) ((f_parabola(x) - f_line(x))/2).^2, x_intersect(1), x_intersect(2));
y_c_below = (1/area_below) * quadgk(@(x) (f_line(x)/2).^2, x_intersect(1), x_intersect(2));
y_c = (area_above * y_c_above + area_below * y_c_below) / (area_above + area_below);

% The centroid is (x_c, y_c)
centroid = [x_c, y_c];
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