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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) $\displaystyle\int_8^b g(x)dx$; $g(x)=~$?

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$\tiny\text{Please don’t shout.}$ All-caps is perceived as shouting by many. –  Brian M. Scott Jan 23 '13 at 1:53
    
Please verify that I correctly interpreted everything when I converted it to $\LaTeX$. –  Brian M. Scott Jan 23 '13 at 1:59
    
Yes, thank you. –  user59255 Jan 23 '13 at 2:01
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2 Answers 2

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).

  • First: Can you see what $\Delta x$ should be?
  • Second: Can you then guess what $f$ could be?
  • Third: If $\Delta x = \frac{b - a}{2}$ where here $a=8$, what would $b$ be?

Now try to do similarly for the first one. Hint: Here you might note that $8 = 2 + 6$.

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(2) $g(x) = \sqrt{x}$, $b=10$.

(1) $f(x) = \sqrt{2+x}$, $b=8$.

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