The following sum
\[
\sqrt{6+\frac{5}{n}} \cdot\left(\frac{5}{n}\right)+\sqrt{6+\frac{10}{n}} \cdot\left(\frac{5}{n}\right)+\ldots+\sqrt{6+\frac{5 n}{n}} \cdot\left(\frac{5}{n}\right)
\]
is a right Riemann sum with $n$ subintervals of equal length for the definite integral
\[
\int_{2}^{b} f(x) d x
\]
where $b=$ and $f(x)=$

It is also a Riemann sum for the definite integral
\[
\int_{6}^{c} g(x) d x
\]
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: \( \boxed{b=11} \), \( \boxed{f(x)=\sqrt{6+x}} \), \( \boxed{c=11} \), \( \boxed{g(x)=\sqrt{6+x}} \).