Which of the following limits is equal to $$ \int_{2}^{5} x^{2} d x $$ (A) $$ \lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(2+\frac{k}{n}\right)^{2} \frac{1}{n} $$ (B) $$ \lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(2+\frac{k}{n}\right)^{2} \frac{3}{n} $$ (C) $$ \lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(2+\frac{3 k}{n}\right)^{2} \frac{1}{n} $$ (D) $$ \lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(2+\frac{3 k}{n}\right)^{2} \frac{3}{n} $$

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To determine which limit is equal to the integral $$ \int_{2}^{5} x^{2} d x $$, we first need to evaluate the integral.

### Step 1: Evaluate the Integral

The integral can be calculated as follows:

$$
\int_{2}^{5} x^{2} d x = \left[ \frac{x^{3}}{3} \right]_{2}^{5}
$$

Calculating the antiderivative at the bounds:

$$
= \frac{5^{3}}{3} - \frac{2^{3}}{3} = \frac{125}{3} - \frac{8}{3} = \frac{117}{3} = 39
$$

### Step 2: Analyze the Limits

Now we will analyze each limit option to see which one equals 39.

#### Option (A)

$$
\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(2+\frac{k}{n}\right)^{2} \frac{1}{n}
$$

This limit represents a Riemann sum for the function $$ f(x) = x^{2} $$ over the interval $$[2, 3]$$ (since $$2 + \frac{k}{n}$$ approaches 2 as $$n$$ approaches infinity). The width of each subinterval is $$\frac{1}{n}$$, which is incorrect for our integral.

#### Option (B)

$$
\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(2+\frac{k}{n}\right)^{2} \frac{3}{n}
$$

This limit represents a Riemann sum for the function $$ f(x) = x^{2} $$ over the interval $$[2, 5]$$ (since $$2 + \frac{k}{n}$$ approaches 5 as $$n$$ approaches infinity). The width of each subinterval is $$\frac{3}{n}$$, which is correct for our integral.

#### Option (C)

$$
\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(2+\frac{3 k}{n}\right)^{2} \frac{1}{n}
$$

This limit represents a Riemann sum for the function $$ f(x) = x^{2} $$ over the interval $$[2, 5]$$ (since $$2 + \frac{3k}{n}$$ approaches 5 as $$n$$ approaches infinity). However, the width of each subinterval is $$\frac{1}{n}$$, which is incorrect for our integral.

#### Option (D)

$$
\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(2+\frac{3 k}{n}\right)^{2} \frac{3}{n}
$$

This limit represents a Riemann sum for the function $$ f(x) = x^{2} $$ over the interval $$[2, 5]$$ (since $$2 + \frac{3k}{n}$$ approaches 5 as $$n$$ approaches infinity). The width of each subinterval is $$\frac{3}{n}$$, which is correct for our integral.

### Conclusion

The correct limit that equals $$ \int_{2}^{5} x^{2} d x $$ is:

**(B)** $$ \lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(2+\frac{k}{n}\right)^{2} \frac{3}{n} $$
Option (B) is the correct limit that equals $$ \int_{2}^{5} x^{2} d x $$.

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