Answer
$$\frac{2}{3}$$
Work Step by Step
Definition 5.4.3
$A=\lim _{n \rightarrow+\infty} \sum_{k=1}^{n} f\left(x_{k}^{*}\right) \Delta x$
$A=\lim _{n \rightarrow+\infty} \sum_{k=1}^{n}\left(\frac{k}{n}\right)^{2} \cdot \frac{1}{n}$
$x_{k}^{*}$ is the left endpoint, $f(x)=x^{2}, \Delta x=\frac{1}{n}$
Rewrite sum in closed form:
\[
A=\lim _{n \rightarrow+\infty} \sum_{k=1}^{n} \frac{(1+2 n)(1+n)}{6 n^{2}}
\]
\[
A=\frac{1}{3}
\]
Evaluate limit
\[
B=-\frac{1}{3}+1=\frac{2}{3}
\]