Question $4[$Numerical Integration$]$ Using a matlab.

The dimensionless temperature of a fluid under steady, fully developed laminar flow in a cylindrical pipe with walls heated electrically is given by $=-4Z-r^2+\frac{r^4}{4}+\frac{7}{24}$ where $r$ is the dimensionless radius. The cup mixing $($bulk$)$ dimensionless temperature is given by

${}_b=\frac{{\displaystyle \underset{r=0}{\overset{1}{}}}(1-r^2)rdr}{{\displaystyle \underset{r=0}{\overset{1}{}}}(1-r^2)rdr}=4{\displaystyle \underset{r=0}{\overset{1}{}}}(1-r^2)rdr.$

$($i$)$ Determine ${}_b$ analytically $($ii$)$ generate values of $$ at $Z=0\mathrm{.}4$ and $r=0,0\mathrm{.}2,0\mathrm{.}4,0\mathrm{.}6,0\mathrm{.}8,1\mathrm{.}0($iii$)$ use a combination of Simpson's $\frac{1}{3}$ and $\frac{3}{8}$ rules to estimate ${}_b$ numerically $($iv$)$ determine the absolute relative true error, $|{}_t|,$ using a Matlab in computer engineering