Answer
$$
320
$$
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{4}{n} \cdot \left(\frac{4 k}{n}+2\right)^{3} $
$x_{k}^{*}$ is the left endpoint, $f(x)=x^{3}, \Delta x=\frac{4}{n}$
Rewrite sum in closed form
\[
A=\lim _{n \rightarrow+\infty} \frac{32\left(4+ 13n+10 n^{2}\right)}{n^{2}}
\]
\[
A=320
\]
