Question

Write Simpson's one-third rule formula in numerical methods.

Answer 1

Simpson's Rule Formula is used to calculate the integral value of any function. It calculates the value of the area under any curve over a given interval by dividing the area into equal parts. It follows the method similar to integration by parts.
In order to integrate any function $f\left(x\right)$ in the interval $(a,b)$, follow the steps given below:

1) Select a value for $n$, which is the number of parts the interval is divided into. Let the value of $n$ be an even number.

2) Calculate the width, $h=\frac{b-a}{n}$

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

4) Consider $y=f\left(x\right)$. Now find the values of $y\left(y_{0}to{y}_n\right)$ for the corresponding $x\left(x_{0}to{x}_n\right)$ values.

5) Substitute all the above found values in the Simpson's Rule Formula to calculate the integral value.

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