The following sum
$$\sqrt{8+\frac2n}\cdot\left(\frac2n\right) + \sqrt{8+\frac4n}\cdot\left(\frac2n\right) + \ldots+ \sqrt{8+\frac{2n}n}\cdot\left(\frac2n\right)$$
is a right Riemann sum for the definite integral.
(1) $\displaystyle\int_6^b f(x)dx$; $f(x)=~$?
It is also a Riemann sum for the definite integral.
(2) $\displaystyle\int_8^b g(x)dx$; $g(x)=~$?
Best Answer
Instead of just giving you the answer, for the second one, you are wanting to compare $$ \sum_{i=1}^{n} f(8 + i\Delta x)\Delta x $$ with $$ \sum_{i=1}^{n} \sqrt{8 + i\frac{2}{n}}\frac{2}{n}. $$ (Noting that this sum is the sum that you have in your question).
Now try to do similarly for the first one. Hint: Here you might note that $8 = 2 + 6$.