In this problem you'll explore how to evaluate the limit of a Riemann sum to calculate $\displaystyle {\int_{{0}}^{{2}}}\sqrt{{{x}}}\ {\left.{d}{x}\right.}$.

Let's partition $\displaystyle {\left[{0},{2}\right]}$ this way: $\displaystyle {P}={\left\lbrace{0},\frac{{{2}\cdot{1}}}{{n}^{{2}}},\frac{{{2}\cdot{4}}}{{n}^{{2}}},\frac{{{2}\cdot{9}}}{{n}^{{2}}},\ldots,\frac{{{2}\cdot{k}^{{2}}}}{{n}^{{2}}},\ldots,\frac{{{2}{\left({n}-{1}\right)}^{{2}}}}{{n}^{{2}}},{2}\right\rbrace}$ 

Answer the following questions:

(a) Write a general expression for $\displaystyle \Delta{x}_{{k}}$ in terms of $\displaystyle {k}$ and $\displaystyle {n}$.

      $\displaystyle \Delta{x}_{{k}}=$        Preview Question 6 Part 1 of 4  

(b) Fill in the blanks for the Riemann sum limit using this partition with $\displaystyle {c}_{{k}}$ as the right endpoint

      of each interval $\displaystyle {\left[{c}_{{{k}-{1}}},{c}_{{k}}\right]}$ and evaluate the limit to compute $\displaystyle {\int_{{0}}^{{2}}}\sqrt{{{x}}}\ {\left.{d}{x}\right.}$.        

$\displaystyle {\int_{{0}}^{{2}}}\sqrt{{{x}}}\ {\left.{d}{x}\right.}=$	$\displaystyle \lim_{{{n}\to\infty}}{\sum_{{{k}={1}}}^{{n}}}$$\displaystyle \Delta{x}_{{k}}$      Preview Question 6 Part 2 of 4
$\displaystyle =$	$\displaystyle \lim_{{{n}\to\infty}}{\sum_{{{k}={1}}}^{{n}}}$       Preview Question 6 Part 3 of 4
$\displaystyle =$	Enter an exact value (no decimals).