Solve using matlab

PROBLEM $4$

The amount of mass transported via a pipe over a period of time can be computed as

$M={\displaystyle \underset{t_1}{\overset{t_2}{}}}Q(t)c(t)dt$

where $M=$ mass $(mg),t_1=$ the initial time $(\{$:min$),t_2=$ the final time $(min),Q(t)=$ flow rate $(m\frac{3}{m}in),$ and $c(t)=$ concentration

$(m\frac{g}{m}3).$ The following functional representations define the temporal variations in flow and concentration:

$Q(t)=9+5co{s}^2(0\mathrm{.}4t)$

$c(t)=5e^{-0\mathrm{.}5t}+2e^{0\mathrm{.}15t}$

Determine the total mass passing through the pipe within the first $15$ seconds. Compare Romberg integral with Simpson's $\frac{3}{8}$ Rule.

Use the following function files to help solve

function EI $=$ IntSimp$38($x$,$y$)$

$\%$ Integral $-$ Simpsons $3/8$ Rule

$\%$ using Least$-$Squares Regression

$\%$Input List

$\%$ x $=$ independent data stream

$\%$ y $=$ dependent data stream

$\%$Output List

$\%$ EI $=$ sum of integral segments

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

Q $=$ floor$(1/3*($N$-1))$;

I $=$ zeros$($Q$,1)$;

for q $=1$:Q

i $=3*($q$-1)+1$;

A $=$ LSRegPoly$($x$($i:i$+3),$y$($i:i$+3),3)$;

I$($q$)=1/4*$A$(1)*($x$($i$+3)^4-$ x$($i$)^4)...$

$+1/3*$A$(2)*($x$($i$+3)^3-$ x$($i$)^3)...$

$+1/2*$A$(3)*($x$($i$+3)^2-$ x$($i$)^2)...$

$+$ A$(4)*($x$($i$+3)-$ x$($i$))$;

end

EI $=$ sum$($I$)$;

if i$+3$ N

EI $=$ EI $+$ IntSimp$13($x$($i$+3$:N$),$y$($i$+3$:N$))$;

end

end

function EI $=$ Romberg$($x$1,$y$1,$x$2,$y$2)$

$\%$ Integral $-$ Newton$-$Cotes Polynomial

$\%$ using Least$-$Squares Regression

$\%$Input List

$\%$ x$1=$ independent data stream #$1$

$\%$ y$1=$ dependent data stream #$1$

$\%$ x$2=$ independent data stream #$2$

$\%$ y$2=$ dependent data stream #$2$

$\%$Output List

$\%$ EI $=$ sum of integral segments

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

if nargin $==2$ && mod$($N$,2)$

x$2=$ x$1(1$:$2$:end$)$;

y$2=$ y$1(1$:$2$:end$)$;

end

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

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

Ia$1=$ NewtonCotes$($x$1,$y$1,1)$;

Ia$2=$ NewtonCotes$($x$2,$y$2,1)$;

$\%$note spacing should be uniform!

EI $=$ Ia$2+($Ia$2-$ Ia$1)/(($dx$1/$dx$2)^2-1)$;

end