Dividing given interval into $n$ equal parts and for every interval taking $\epsilon$ as an midpoint of an interval find integral sum of a function and compute it.
$y=1+x$, $x \in[-1,4]$
$P$ is given splitting of an inteval.
$\sigma(P,\epsilon)$ = $\sum_{i=0}^{n-1}f(\epsilon)\triangle x_i$
Because length of an interval is $5$
$\triangle x_i=\frac{5}{n}$ and $\epsilon_i= \frac{x_{i+1}-x_i}{2}$
$\sum_{i=0}^{n-1}(1+\epsilon_i)\triangle x_i=\sum_{}(1+\frac{10}{n}){\frac{5}{n}}$ which doesn't given answer in the book. Answer is $12.5.$
We have that
$$\epsilon_i=\frac{x_{i}+x_{i+1}}{2}=\frac12\left(-1+5\frac i n-1+5\frac {i+1} n\right)=-1+\frac 5 2\frac{2i+1}n \implies f(\epsilon_i)=\frac 5 2\frac{2i+1}n$$
and therefore
$$\sum_{i=0}^{n-1}\frac52\frac{2i+1}n\triangle x_i=\frac{25}{2n^2}\sum_{i=0}^{n-1}(2i+1)=\frac {25}{ 2n^2}n^2=\frac {25}{ 2}$$