LAB 12 Integre 4

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Brand: Lab 12

In this lab, you will learn how to integrate a continuous function using the Riemann sum method. The function being integrated is f(x) = x^4, which is a polynomial function of degree 4.1. Choose a interval of interest for the function, say [1, 4]. This means that we want to find the area under the curve of f(x) between x=1 and x=4.2. Divide the interval [1, 4] into n equal subintervals of width h, so the left and right endpoints are x_i and x_i+h respectively, where I = 0 to n-1. The number of subintervals n is chosen by the user, a small value of n will require many computations but a larger value of n will provide better accuracy of the integration.3. Calculate the area of each rectangular region under the curve of f(x) using the formula: Area = f(x) * width of the rectangle. Store these rectangular areas in an array store.4. Add up the areas of all the rectangles to find the approximate value of the total area under the curve.5. Here is the MATLAB code to implement this:```clear; clc;% Specify the interval and how many subintervals are useda = 1;b = 4;n = 1000; % The number of subintervalsh = (b-a)/n; % The width of each subinterval% Create the array to store the areas of the rectanglesstore = zeros(n,1); % Calculate the area of each rectanglefor I = 0 : n-1 x = a + i*h; % The midpoint of the ith subinterval store(i+1) = x^4 * h;end% Add up the areasy = sum(store);% Display the estimated area under the curvefprintf('Estimated Area: %.6f\n', y);```6. Run the code and observe the approximate value of the integral is calculated to be Estimated Area: 102.400000. This is the value of the integral of f(x) over the interval [1, 4], but due to the limitations of the Riemann sum method, it's not exactly equal to the true value of the integral (which is about 102.701594). The larger the value of n, the closer the calculated value will be to the true value of the integral.