Imagine dividing $0,$ into $N=8$ subintervals, with endpoints

$x_0=0,x_1=\frac{}{8},$dots,$x_i=\frac{i}{8},$dots,$x_8=\frac{8}{8},$

and evaluating the numbers $f_i=f(x_i)$ for $i=0,1,$dots,$8.$ For $f(x)$ as shown in Figure $1,$ the $9$ numbers $f_i,$ in order, are approximately

$0\mathrm{.}0,409\mathrm{.}3,149\mathrm{.}8,-53\mathrm{.}9,75\mathrm{.}0,19\mathrm{.}3,-52\mathrm{.}2,50\mathrm{.}5,0\mathrm{.}0$

$($a$)(1$ mark$)******$ Use all of these values to calculate Trapezoidal$-$rule approximations for the integrals $B_1(f),$dots,$B_5(f).($Call these $B_n^{\text{t}\text{r}\text{a}\text{p}}(f).)$ Report each answer with $3$ decimal places.

Your writeup for this problem should describe your method for computing the numbers $B_n^{\text{t}\text{r}\text{a}\text{p}}(f)$ in enough detail that a patient reader could reproduce your work. Show documentation appropriate for whatever approach you used: a code listing $($any language is acceptable$),$ a spreadsheet sample, or selected screenshots, etc.

The next question applies the ideas from Question $3$ at scale. It involves rather large numbers $N$ and $R,$ so computer assistance in evaluating the individual Trapezoidal$-$Rule approximations will be essential. You will need to apply the Trapezoidal Rule to $R+1$ different integrals, so a systematic computer$-$assisted approach