Answer
\[
\frac{39}{4}
\]
Work Step by Step
Definition 5.4.3
$A=\lim _{n \rightarrow+\infty} \sum_{k=1}^{n} f\left(x_{k}^{*}\right) \Delta x$
$A=\lim _{n \rightarrow+\infty} \sum_{k=1}^{n}\left(4-\frac{9 k^{2}}{4 n^{2}}\right) \cdot \frac{3}{n}$
$x_{k}^{*}$ is the left endpoint, $f(x)=-\frac{1}{4} x^{2}+4, \Delta x=\frac{3}{n}$
\[
A=\lim _{n \rightarrow+\infty} \frac{\left(-3-9 n+26 n^{2}\right)3}{8 n^{2}}
\]
Rewrite sum in closed form
\[
A=\frac{39}{4}=9.75
\]
