How to plot graph regarding the definite integral by using Midpoint rule for the function 𝑦 =x^1/3?

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L
L 2024-3-6
评论： L 2024-3-7
This is what I have done and I'm not sure is it correct.
I don't know how to continue to plot the graph of it.
y = @(x) nthroot(x,3); %Function to integrate
a = 0; b = 2; %Interval where a is the lower boundary and b is the upper boundary.
n = 10; %Number of subintervals
dx = (b-a/n); %Width of each rectangle
mpr = 0;
for i = a:dx:b
mpr = mpr + f(i+dx/2);
end
I = dx*mpr;
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Hassaan
Hassaan 2024-3-6
编辑：Hassaan 2024-3-6
Ran in:
% Define the function to integrate using a function handle
f = @(x) nthroot(x,3);
% Define the interval and the number of subintervals
a = 0; b = 2; % Interval [a, b]
n = 10; % Number of subintervals
% Calculate the width of each subinterval
dx = (b - a) / n;
% Initialize the midpoint sum
mpr = 0;
% Calculate the sum for the Midpoint Rule
for i = 1:n
xi = a + (i-0.5)*dx; % Midpoint of the i-th subinterval
mpr = mpr + f(xi); % Add the value at the midpoint
end
% Calculate the integral approximation
I = dx * mpr;
% Display the result
fprintf('The approximate value of the integral is: %f\n', I);
The approximate value of the integral is: 1.896224
% Plotting the function and the rectangles
x_vals = linspace(a, b, 1000); % Generate 1000 points between a and b
y_vals = arrayfun(f, x_vals); % Evaluate the function at each x value
% Plot the function
plot(x_vals, y_vals, 'b-', 'LineWidth', 1.5);
hold on; % Hold on to the current plot
% Plot each rectangle
for i = 1:n
xi = a + (i-0.5)*dx; % Midpoint of the i-th subinterval
% Plot the rectangle
rect_x = [xi - dx/2, xi - dx/2, xi + dx/2, xi + dx/2];
rect_y = [0, f(xi), f(xi), 0];
patch(rect_x, rect_y, 'r', 'EdgeColor', 'r', 'FaceAlpha', 0.1);
end
% Formatting the plot
xlabel('x');
ylabel('y');
title('Midpoint Rule Approximation');
legend('y = nthroot(x, 3)', 'Midpoint rectangles');
hold off; % Release the plot hold
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It's important to note that the advice and code are based on limited information and meant for educational purposes. Users should verify and adapt the code to their specific needs, ensuring compatibility and adherence to ethical standards.
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