Answer
(b) $\lim _{\max \Delta x_{k} \rightarrow 0} \sum_{k=1}^{n}\Delta x_{k} \frac{x_{k}^{*}}{1+x_{k}^{*}} $
(a) $\lim _{\max } \Delta x_{k} \rightarrow 0 \sum_{k=1}^{n} \Delta x_{k} 2 x_{k}^{*} $
Work Step by Step
Replace the integration with the limit of the sum, $ d x $ with $\Delta x_{k}$, and $x$ with $x_{k}^{*}$.
$(\mathrm{a}) \lim _{\max \Delta x_{k} \rightarrow 0} \sum_{k=1}^{n} \Delta x_{k} 2 x_{k}^{*} $
$b=2$ and $a=1$
(b) $\lim _{\max \Delta x_{k} \rightarrow 0} \sum_{k=1}^{n} \Delta x_{k} \frac{x_{k}^{*}}{x_{k}^{*}+1}$
$b=1$ and $a=0$
