Part $1$: Multiple Segment Simpson's $1/3$ Rule

Aside from applying the trapezoidal rule with finer segmentation, another way to obtain a more

accurate estimate of an integral is to use higher$-$order polynomials to connect the points.

Simpson's $\frac{1}{3}$ rule results when a second$-$order interpolating polynomial is used. And it can be

improved by dividing the integration interval into a number of segments of equal width $($Multiple

Segment Simpson's $1/3$ Rule$)$

Write a matlab function$/$code that does calculates the integral of a function in a given interval

with multiple$-$segment Simpson's $\frac{1}{3}$ rule, where the approximating the integral is as follows:

$I$~$=(b-a)\frac{f(x_0)+4{\displaystyle \underset{i=1}{\overset{n-1}{}}},3,5f(x_i)+2{\displaystyle \underset{j=2}{\overset{n-2}{}}},4,6f(x_j)+f(x_n)}{3n}$

Generate a matlab function$/$code and call it: simpson.m

An algorithm $($for a matlab function$)$ you use may start as follows $($you need to complete the

remaining part according to Equation $1).$ Because of the need for three points for application of

$\frac{1}{3}$ rule each segment, the method is limited to odd number of points $($even number of

segments$).$

sinput $-$all the constants in your function

sinput $-f$ is the integrand

$x0$ and $xn$ are lower and upper limits of integration

$n$ is the number of segments

Output $-S$ is the simpson rule sum.

sl$=0$; in inial sum for the odd terms set to be zero

$s2=0$; $$ initial sum for the even terms set to be zero

s you need to write two "for loop" for the sum of the odd and even terms

COMPLETE the program script on your own