numerical integration in matlab

There are different ways to perform numerical integration in Matlab, depending on the nature of the integrand, the accuracy required, and the available computational resources. Here are some examples using different functions in Matlab:

1. Quad function: this function provides adaptive numerical integration of functions over a finite interval. The syntax is Q = quad(fun,a,b), where fun is the handle to the integrand, and a and b are the lower and upper limits of integration, respectively. Here's an example:

main.m
% Define the integrand
fun = @(x) sin(x.^2);

% Define the limits of integration
a = 0;
b = 1;

% Compute the integral using quad
Q = quad(fun,a,b);

% Display the result
fprintf('The integral of sin(x^2) from %g to %g is %g.\n',a,b,Q);
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2. Integral function: this function performs numerical integration of functions over a finite or infinite interval. The syntax is Q = integral(fun,a,b), where fun is the handle to the integrand, and a and b are the lower and upper limits of integration, respectively. Here's an example:

main.m
% Define the integrand
fun = @(x) exp(-x.^2);

% Define the limits of integration (from -inf to inf)
a = -inf;
b = inf;

% Compute the integral using integral
Q = integral(fun,a,b);

% Display the result
fprintf('The integral of exp(-x^2) from -inf to inf is %g.\n',Q);
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3. Trapezoidal rule: this is a simple method for numerical integration of sampled data. The syntax for computing the integral of a vector of data y with respect to x using the trapezoidal rule is Q = trapz(x,y). Here's an example:

main.m
% Define some sampled data
x = linspace(0,1,101);
y = sin(x.^2);

% Compute the integral of y with respect to x using trapz
Q = trapz(x,y);

% Display the result
fprintf('The integral of sin(x^2) from %g to %g is approximately %g.\n',x(1),x(end),Q);
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4. Cumulative trapezoidal rule: this method is similar to the trapezoidal rule, but it computes the cumulative integral of the sampled data. The syntax for computing the cumulative integral of a vector of data y with respect to x using the cumulative trapezoidal rule is Q = cumtrapz(x,y). Here's an example:

main.m
% Define some sampled data
x = linspace(0,1,101);
y = sin(x.^2);

% Compute the cumulative integral of y with respect to x using cumtrapz
Q = cumtrapz(x,y);

% Plot the cumulative integral
plot(x,Q);
xlabel('x');
ylabel('Cumulative integral of sin(x^2)');
title('Cumulative trapezoidal rule');
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These are just some examples of how to perform numerical integration in Matlab. There are many other methods and functions available, depending on the specific requirements of the problem.