Question

In numerical methods, write the formula for Simpson's one-third rule.

Answer 1

To integrate a function $f\left(x\right)$ in the interval $(a,b)$ we can use Simpson's one third rule.

Divide the interval into $n$ parts. Let the value of $n$ is even.

Then width $h=\frac{b-a}{n}$.

Calculate the values of $x_0$ to $x_n$ as $x_0=a,x_1=x_0+h,....,x_{n-1}=x_{n-2}+h,x_n=b$.

Consider $y=f\left(x\right)$. Now find the value of $y(y_0$ to $y_n)$ for the corresponding $x(x_0$ to $x_n)$ values.

Now, substitute all the values in the Simpson's one third rule.

The formula for Simpson's one third rule is :

${\int }_a^{b}f\left(x\right)dx=\frac{h}{3}[(y_0+y_n)+4(y_1+y_3+...+y_{n-1})+2(y_2+y_4+...+y_{n-2})]$