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

$1$

$3$

$$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

$1$

$3$

$$rule$,$ where the approximating the integral is as follows:

$$

~

$=$

$($

$$

$-$

$$

$)$

$$

$($

$$

$0$

$)$

$+$

$4$

$$

$$

$=$

$1$

$$

$-$

$1$

$,$

$3$

$,$

$5$

$$

$($

$$

$$

$)$

$+$

$2$

$$

$$

$=$

$2$

$$

$-$

$2$

$,$

$4$

$,$

$6$

$$

$($

$$

$$

$)$

$+$

$$

$($

$$

$$

$)$

$3$

$$

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

$1$

$3$

$$rule each segment, the method is limited to odd number of points

$($

even number of

segments

$)$

$.$

sinput

$-$

all the constants in your function

sinput

$-$

$$

$$is the integrand

$$

$0$

$$and

$$

$$

$$are lower and upper limits of integration

$$

$$is the number of segments

Output

$-$

$$

$$is the simpson rule sum.

sl

$=$

$0$

; in inial sum for the odd terms set to be zero

$$

$2$

$=$

$0$

;

$$

$$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