Simpson's rule

Simpson's rule states that the integral of a function over a small interval $dx$ is,

$$\int \limits_{x_0-dx}^{x_0} f(x) dx \approx \frac{dx}{6}\left(f(x_0-dx) + 4f\left(\frac{x_0-dx/2}{2}\right)+f(x_0)\right).$$

The integral, $$ I = \int \limits_a^b f(x) dx, $$

can then be performed by dividing $(b-a)$ into a number of small intervals and summing the results. The code snipet below performs this function.

<script> //define the function f function f(x) { y = xx; return y; } // the function f will be integrated from a to b a = 0; b= 2; N_interval= 100; // the number of intervals dx = (b-a)/N_interval; //step size Integral = 0; //Initialize the integral for (i=1; i < N_interval+1; i++) { x = a + idx Integral += (dx/6)(f(x-dx)+4f(x-dx/2)+f(x)); } document.write(" Integral = "+Integral+"<br \/>"); </script>

This is not numerically efficient because the function $f(x)$ is called twice for the same value of $x$ in every interval.