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

Hint:
Simpson’s rule formula is used to calculate the integral value of any function. It calculates the value of the area under the curve over a given interval by dividing the area into equal parts. It follows the method similar to integration by parts.

Complete step by step solution:
Step – 1 : In order to integrate any function $f(x)$ in the interval $\left( {a,b} \right)$ follow the steps given below.
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.
Step – 2 : Calculate the width, is denoted by$h$ using formula for find width $\left( h \right) = \dfrac{{b - g}}{n}$
Where a and b denoted the interval $\left( {a,b} \right)$
Step – 3 : Calculate the value of \[{x_0}\] to \[{x_n}\] as \[{x_0} = 9\], \[{x_1} = {x_0} + h,.......{x_{n - 1}} = {x_{n - 2}} + h,{x_n} = b\]

Step – 4 : Substitute all the above found value, in the Simpson’s rule formula to calculate the integral value.
\[\int\limits_a^b {f(x)dx = \dfrac{h}{3}} \left[ {\left( {{y_0} + {y_1}} \right) + y\left( {{y_1} + {y_3} + ... + {y_{n - 1}}} \right) + 2\left( {{y_2} + {y_4} + ... + {y_{n - 2}}} \right)} \right]\]

Note: Integration is the process of measuring the area under a function plotted on a graph. Sometimes, the evaluation of expression involving these integrals can become daunting, if not indeterminate.
Here, we will discuss the Simpson’s $\dfrac{1}{3}$ rule of approximating integrals of the form
\[I = \int\limits_a^b {f(x)dx} \] Where, \[f(x)\] is called integral
a = lower limit of integration
b = upper limit of integration
Simpson’s $\dfrac{1}{3}$ rule is an extension of Trapezoidal rule where the integral is approximated by a second order polynomial.

Note: