Answer
(b) $\lim _{\max \Delta x_{k} \rightarrow 0} \sum_{k=1}^{n} \Delta x_{k} \left(\cos x_{k}^{*}+1\right)$
(a) $\lim _{\max \Delta x_{k} \rightarrow 0} \sum_{k=1}^{n} \Delta x_{k} \sqrt{x_{k}^{*}} $
Work Step by Step
Replace the integration with the limit of the sum, $ d x $ with $\Delta x_{k}$, and $x$ with $x_{k}^{*}$.
(a) $\lim _{\max \Delta x_{k} \rightarrow 0} \sum_{k=1}^{n} \Delta x_{k} \sqrt{x_{k}^{*}} $
Boundaries interval
$b=2$ and $a=1$
Replace the integration with the limit of the sum, $ d x $ with $\Delta x_{k}$, and $x$ with $x_{k}^{*}$.
(b) $\lim _{\max } \Delta x_{k} \rightarrow 0 \sum_{k=1}^{n} \Delta x_{k} \left(\cos x_{k}^{*}+1\right) $
Boundaries interval
\[
b=\frac{\pi}{2} \text { and } a=-\frac{\pi}{2}
\]