Consider the function $f(x)=-\frac{x^2}{2}-1.$

$...$

In this problem you will calculate ${\displaystyle \underset{0}{\overset{4}{}}}(-\frac{x^2}{2}-1)dx$ by using the definition:

${\displaystyle \underset{a}{\overset{b}{}}}f(x)dx=\underset{n}{lim}[{\displaystyle \underset{i=1}{\overset{n}{}}}f(x_i)x]$

The summation inside the brackets is $R_n$ which is the Riemann sum where the sample points are chosen to be the right$-$hand endpoints of each sub$-$interval.

Calculate $R_n$ for $f(x)=-\frac{x^2}{2}-1$ on the interval $[0,4]$ and write your answer as a function of n without any summation signs.

$R_n=$

lim $R_n=$

n $->$