Answer
$$
22
$$
Work Step by Step
Definition 5.4.3
$A=\lim _{n \rightarrow+\infty} \sum_{k=1}^{n} f\left(x_{k}^{*}\right) \Delta x$
$x_{k}^{*}$ is the left endpoint, so:
$A=\lim _{n \rightarrow+\infty} \sum_{k=1}^{n}\left(1-\left(-3+\frac{2 k}{n}\right)^{3}\right) \cdot \frac{2}{n}$
$f(x)=-x^{3}+1, \Delta x=\frac{2}{n}$
Rewrite sum in closed form:
\[
A=\lim _{n \rightarrow+\infty} \frac{22 n^{2}-26 n+8}{n^{2}}
\]
$A=22$