Problem $1(20$ points, Core Course Outcome $\backslash$#$6)/$ Mat labGrader

Develop a Matlab function mysimpson$38$ that calculates $\backslash ($ I$=\backslash$int y$($x$)$ d x $\backslash )$ using the composite Simpson's $\backslash (3/8\backslash )$ method for a set of discrete data points $\backslash (\backslash$left$($x$\_\{$i$\},$ y$\_\{$i$\}\backslash$right$)\backslash )$ that are equally spaced. If the number of subintervals is not divisible by $3$ because there is $1$ extra subinterval, calculate the integral of the last $4$ subintervals using Simpson's $\backslash (1/3\backslash )$ method. If there are $2$ extra subintervals, calculate the integral for the last $2$ subintervals using Simpson's $\backslash (1/3\backslash )$ method.

If the number of subintervals is zero $(\backslash (=1\backslash )$ data point only$),$ the function shall return a value of zero for the integral. If the number of subintervals is one $(\backslash (=2\backslash )$ data points only$),$ the function shall use the trapezoidal method on the single interval. If the vectors $\backslash ($ x $\backslash )$ and $\backslash ($ y $\backslash )$ are not of equal length, the function shall execute Matlab's function error$("...")$ with an appropriate error message.

As input the function shall take the column vectors x and y that contain the data points. As output the function shall give the calculated integral I. You may calluse any function developed in recitation, homework, and$/$or exams.

Required submission:

$\backslash (\backslash$square $\backslash )$ well commented function source code submitted to Matlab Grader using the Canvas link for Exam $6-$ Problem $1$

Here is the function for the trapezoidal method:

function $[$I$]=$ myTrapezoidal$($x$,$y$)$

$\%$ calculates the integral y$($x$)$dx for the given data points using the

$\%$ composite trapezoidal method

$\%$ Inputs: x$,$y $=$ data points

$\%$ Outputs: I $=$ integral

$\%$ checks for equal number of entries in x and y

if length$($x$)$ ~$=$ length$($y$)$

error$("$Vectors x and y must have the same number of entries."$)$;

end

$\%$ if only one data point is in x and y$,$ the integral is zero

if length$($x$)==1$

I $=0$;

return

end

$\%$ initializes runnign sum of $1$

I $=0$;

$\%$ number of intervals N is the number of points minus $1$

N $=$ length$($x$)-1$;

$\%$ evaluates the composite trapezoidal method with a loop

for i $=1$:N

I $=$ I$+1/2*($y$($i$)+$y$($i$+1))*($x$($i$+1)-$x$($i$))$;

end

end

Here is the function for Simpson's $1/3$ Method:

function $[$I$]=$ mySimpson$13($x$,$ y$)$

$\%$ evaluate integral y$($x$)$dx using Simpson's $1/3$ composite method for the

$\%$ data points given in x and y

$\%$ Inputs: x$,$y $=$ data points

$\%$ Outputs: I $=$ integral evaluated using Simpson's composite $1/3$ method

$\%$ check that input vectors x and y have the same length

if length$($x$)$ ~$=$ length$($y$)$

error$("$Vectors x and y must have the same length"$)$;

end

$\%$ if only one data point in x and y$,$ the integral is zero

if length$($x$)==1$

I $=0$;

return;

end

$\%$ if exactly two data points, calculate using the trapezoidal method

if length$($x$)==2$

I $=1/2*($y$(1)+$ y$(2))*($x$(2)-$ x$(1))$;

return; $\%$ Return immediately after calculating the trapezoidal area

end

$\%$ Number of intervals N is the number of points minus $1$

N $=$ length$($x$)-1$;

$\%$ Initialize integral

I $=0$;

$\%$ For an odd number of intervals, use trapezoidal method for the last interval

if mod$($N$,2)==1$

I $=1/2*($y$($N$)+$ y$($N $+1))*($x$($N $+1)-$ x$($N$))$; $\%$ trapezoidal for the last interval

N $=$ N $-1$; $\%$ reduce N for Simpson's method

end

$\%$ Use Simpson's $1/3$ method for the remaining intervals

h $=$ x$(2)-$ x$(1)$;

I $=$ I $+$ h$/3*($y$(1)+4*$ sum$($y$(2$:$2$:N$+1))+2*$ sum$($y$(3$:$2$:N$))+$ y$($N $+1))$;

end