Answer
$$A=\frac{39}{4}$$
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:
$A=\lim _{n \rightarrow+\infty} \sum_{k=1}^{n}\left(-\frac{(-1+k)^{2}9}{4 n^{2}}+4\right) \cdot \frac{3}{n}$
$f(x)=-\frac{1}{4} x^{2}+4, \frac{3}{n}=\Delta x$
Rewrite sum in closed form:
\[
A=\lim _{n \rightarrow+\infty} \frac{\left(-3+9 n+26 n^{2}\right)3}{8 n^{2}}
\]
$A=\frac{39}{4}$
