Answer
$$\frac{15}{4}$$
Work Step by Step
Definition 5.4 .3
$A=\lim _{n \rightarrow+\infty} \sum_{k=1}^{n} f\left(x_{k}^{*}\right) \Delta x$
$A=\lim _{n \rightarrow+\infty} \sum_{k=1}^{n} \frac{\frac{3 k}{n}+1}{2} \cdot \frac{3}{n}$
$x_{k}^{*}$ is the left endpoint, $f(x)=x / 2, \Delta x=\frac{3}{n}$
$A=\lim _{n \rightarrow+\infty} \frac{3}{2 n} \sum_{k=1}^{n} \frac{9}{2 n^{2}}+1 \sum_{k=1}^{n} k$
$=\lim _{n \rightarrow+\infty}\left(\frac{3 n}{2 n}+\frac{(1+n)9 n}{4 n^{2}}\right)$
$=\lim _{n \rightarrow+\infty}\left(\frac{9 n+15 n^{2}}{4 n^{2}}\right)$
Rewrite sum in closed form
\[
\frac{15}{4}=A
\]
