how do i implement simpson's integration rule for a function f(x) in matlab

To implement Simpson's Integration Rule for a function f(x) in Matlab, you can follow these steps:

1. Define the function f(x) that you want to integrate. For example, if you want to integrate f(x) = x^2 for x = [0,1], you can define it as:

main.m
function y = f(x)
y = x.^2;
end
32 chars
4 lines

2. Define the limits of integration a and b, and the number of intervals n. For example, to integrate f(x) over [0,1] using 10 intervals, you can define:

main.m
a = 0;
b = 1;
n = 10;
22 chars
4 lines

3. Compute the interval width h = (b-a)/n, and the x values of the interval midpoints x_i = a+i*h.

main.m
h = (b-a)/n;
x = linspace(a,b,n+1);
x_mid = x(1:end-1) + h/2;
62 chars
4 lines

4. Compute the values of f(x) at the interval midpoints.

main.m
f_mid = f(x_mid);
18 chars
2 lines

5. Use the Simpson's Integration Rule formula to compute the integral.

main.m
integral = (h/6)*(f(a) + 4*sum(f_mid) + 2*sum(f(x_mid+(h/2))) + f(b));
71 chars
2 lines

The integral variable now contains the value of the integral of f(x) over [a,b] using Simpson's Integration Rule.

Here's the complete implementation for our example function f(x) = x^2, over [0,1] with 10 intervals:

main.m
function y = f(x)
y = x.^2;
end

a = 0;
b = 1;
n = 10;

h = (b-a)/n;
x = linspace(a,b,n+1);
x_mid = x(1:end-1) + h/2;

f_mid = f(x_mid);

integral = (h/6)*(f(a) + 4*sum(f_mid) + 2*sum(f(x_mid+(h/2))) + f(b));

disp(integral)
225 chars
18 lines

This will output: 0.3333, which is the expected value of the integral of x^2 over [0,1].