The following sum
$$\sqrt{6+\frac{5}{n}}\cdot \left(\frac{5}{n}\right)+\sqrt{6+\frac{10}{n}}\cdot \left(\frac{5}{n}\right)+\dots +\sqrt{6+\frac{5n}{n}}\cdot \left(\frac{5}{n}\right)$$
is a right Riemann sum with $n$ subintervals of equal length for the definite integral
$${\int }_2^{b}f(x)dx$$
where $b=$ and $f(x)=$

It is also a Riemann sum for the definite integral
$${\int }_6^{c}g(x)dx$$
where $c=$ and $g(x)=$
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Step 1 ：The given sum is a Riemann sum, which is a method for approximating the total area underneath a curve on a graph, also known as an integral. The sum is in the form of $\sqrt{6+\frac{5i}{n}}\cdot \left(\frac{5}{n}\right)$, where $i$ ranges from 1 to $n$. This can be interpreted as the sum of the areas of $n$ rectangles, where the height of each rectangle is given by the function value at the right endpoint of the subinterval, and the width of each rectangle is $\frac{5}{n}$.

Step 2 ：The function $f(x)$ is the integrand, which is the function being integrated. In this case, the integrand is $\sqrt{6+x}$, because this is the function that is being evaluated at the right endpoint of each subinterval.

Step 3 ：The lower limit of integration is the value at which the first rectangle starts, which is when $i=1$. Substituting $i=1$ into $\frac{5i}{n}$ gives $\frac{5}{n}$, so the lower limit of integration is $6+\frac{5}{n}=6+5=11$. The upper limit of integration is the value at which the last rectangle ends, which is when $i=n$. Substituting $i=n$ into $\frac{5i}{n}$ gives $5$, so the upper limit of integration is $6+5=11$.

Step 4 ：Therefore, the definite integral that the given sum represents is ${\int }_2^{11}\sqrt{6+x}dx$.

Step 5 ：The function $g(x)$ and the limits of integration for the second definite integral can be found in a similar way. The function $g(x)$ is the same as $f(x)$, because the integrand is the same.

Step 6 ：Therefore, the second definite integral that the given sum represents is ${\int }_6^{11}\sqrt{6+x}dx$.

Step 7 ：Final Answer: $\boxed{{\displaystyle b=11}}$, $\boxed{{\displaystyle f(x)=\sqrt{6+x}}}$, $\boxed{{\displaystyle c=11}}$, $\boxed{{\displaystyle g(x)=\sqrt{6+x}}}$.