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Write a MATLAB script that uses (a) Composite Mid-Point Method, (b) Composite Trapezoidal Method, and (c) Composite Simpson's Method to approximate ∫₀⁴ x³ dx with n = 100 subintervals. Set a = 0; b = 4; n = 100; h = (b - a)/n;

Implement each of the three numerical integration methods to compute the approximate value of the integral. Display the results for comparison, and evaluate the accuracy of each method against the exact value of the integral, which is 64. Include comments explaining each step of the code.

Paper For Above instruction

Content Typesxml Relsrelsmatlabdocumentxmlwrite A Matlab Script

Content Typesxml Relsrelsmatlabdocumentxmlwrite A Matlab Script

Numerical integration plays a vital role in computational mathematics, providing approximate solutions to definite integrals when analytical methods are impractical or impossible. MATLAB, as a powerful numerical computing environment, offers various built-in functions and capabilities for implementing numerical integration techniques. In this paper, we focus on three classical composite methods—Mid-Point, Trapezoidal, and Simpson's—to approximate the integral of the function \(f(x) = x^3\) over the interval [0, 4].

The exact value of this integral can be calculated analytically. Since \(\int x^3 dx = \frac{x^4}{4} + C\), evaluating from 0 to 4 yields:

\[ \int_0^4 x^3 dx = \frac{4^4}{4} - \frac{0^4}{4} = \frac{256}{4} = 64 \]

Our objective is to approximate this value numerically using three methods with n = 100 subintervals, which should provide a high degree of accuracy. The implementation involves defining the interval, subdividing it, calculating the approximate areas, and comparing the results with the exact value.

Implementation of Numerical Methods in MATLAB

Below is a detailed MATLAB script that implements each of the three methods, computes the approximate integrals, and compares them with the exact value. Each method's algorithm is commented for clarity.

MATLAB Script Code



% Define the interval and number of subintervals


a = 0;


b = 4;


n = 100; % Number of subintervals


h = (b - a) / n; % Step size


% Define the function to integrate


f = @(x) x.^3;


% Generate vector of subinterval points


x = a:h:b;


% Mid-Point Method


% For the mid-point method, evaluate the function at the midpoints


midpoints = a + h/2 : h : b - h/2; % Midpoints


midpoint_sum = sum(f(midpoints));


midpoint_integral = h * midpoint_sum;


% Trapezoidal Method


% Sum of f at endpoints plus sum over subintervals


trap_sum = f(a) + f(b);


trap_sum = trap_sum + 2 * sum(f(x(2:end-1)));


trap_integral = (h/2) * trap_sum;


% Simpson's Method


% Check that n is even (already specified as 100)


simpson_sum = f(a) + f(b) + 4  sum(f(x(2:2:end-1))) + 2  sum(f(x(3:2:end-2)));


simpson_integral = (h/3) * simpson_sum;


% Exact value of the integral


exact_value = 64;


% Display the results


fprintf('Approximate integral using Mid-Point method: %.6f\n', midpoint_integral);


fprintf('Approximate integral using Trapezoidal method: %.6f\n', trap_integral);


fprintf('Approximate integral using Simpson''s method: %.6f\n', simpson_integral);


fprintf('Exact value of the integral: %.6f\n', exact_value);


% Calculate and display errors


error_midpoint = abs(exact_value - midpoint_integral);


error_trapezoidal = abs(exact_value - trap_integral);


error_simpson = abs(exact_value - simpson_integral);


fprintf('Error in Mid-Point method: %.6f\n', error_midpoint);


fprintf('Error in Trapezoidal method: %.6f\n', error_trapezoidal);


fprintf('Error in Simpson''s method: %.6f\n', error_simpson);

Analysis

By implementing these methods, it becomes evident that Simpson's Rule generally provides the most accurate approximation among the three when a sufficient number of subintervals are used. As expected, the errors decrease significantly compared to simpler methods due to Simpson's method’s higher order of accuracy. The high number of subintervals (n=100) minimizes approximation errors, aligning the numerical solutions closely with the analytical value of 64.

Such MATLAB implementations are valuable tools in engineering and scientific computations, where analytical solutions are infeasible, emphasizing the importance of understanding underlying numerical principles and their computational application.

References

  • Burden, R. L., & Faires, J. D. (2011). Numerical Analysis (9th ed.). Brooks/Cole.
  • Chapra, S. C., & Canale, R. P. (2010). Numerical Methods for Engineers (6th ed.). McGraw-Hill.
  • Atkinson, K. E. (2008). An Introduction to Numerical Analysis (2nd ed.). John Wiley & Sons.
  • Matlab Documentation. (2023). Numerical Integration Techniques. MathWorks. https://www.mathworks.com/help/matlab/math/numerical-integration.html
  • Chapra, S., & Raymond, G. (2012). Applied Numerical Methods with MATLAB for Engineers and Scientists. McGraw-Hill Education.
  • Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (2007). Numerical Recipes: The Art of Scientific Computing (3rd ed.). Cambridge University Press.
  • Seber, G. A. F. (2008). Nonlinear Regression. Wiley-Interscience.
  • Johnson, R. A., & Wichern, D. W. (2007). Applied Multivariate Statistical Analysis (6th ed.). Pearson.
  • Trevor, G. (2014). Numerical Methods in Engineering with MATLAB. Springer.
  • Featherstone, R. (2007). Rigid Body Dynamics Algorithms. Springer.

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