Answer
$$\frac{2}{3}$$
Work Step by Step
Definition 5.4.3
$A=\lim _{n \rightarrow+\infty} \sum_{k=1}^{n} f\left(x_{k}^{*}\right) \Delta x$
$A=\lim _{n \rightarrow+\infty} \sum_{k=1}^{n}\left(\frac{-1+4 k^{2}}{n^{2}}\right) \cdot \frac{2}{n}$
$x_{k}^{*}$ is the left endpoint, $f(x)=-1+x^{2}, \Delta x=\frac{2}{n}$
Rewrite sum in closed form:
\[
A=\lim _{n \rightarrow+\infty} \frac{\left(2+6n+n^{2}\right)2}{3 n^{2}}
\]
$A=\frac{2}{3}$
