Answer
$$
18
$$
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{3}{n} \cdot \left(-\frac{9 k^{2}+9}{n^{2}}\right) $
$x_{k}^{*}$ is the left endpoint, $f(x)=-x^{2}+9, \frac{3}{n}=\Delta x$
$A=\lim _{n \rightarrow+\infty} \frac{27}{n} \sum_{k=1}^{n} -\frac{27}{n^{3}+1} \sum_{k=1}^{n} k^{2}$
$=\lim _{n \rightarrow+\infty}\left(-\frac{27 n(1+n)(1+2 n)}{\frac{27 n}{n}+6 n^{3}}\right)$
$=\lim _{n \rightarrow+\infty}\left(\frac{-27 n(1+n)(1+2 n)+162 n^{3}}{6 n^{3}}\right)$
Rewrite sum in closed form
\[
A=-\frac{54}{6}+\frac{162}{6}=18
\]
