We are given a Riemann sum:

$\lim_{n \to \infty} \sum_{k=1}^{n} \frac{10k}{n} \left( \sqrt{1 + \frac{5k}{n}} \right) \cdot \frac{5}{n}$

To interpret this as a definite integral, recall that the Riemann sum represents an integral of the form:

$\int_{a}^{b} f(x) \, dx$

where:

• $\frac{1}{n}$ represents a small width $\Delta x$,
• $k$ represents a sample point, which approximates the value of the function at that point.

Step 1: Identify the form of the Riemann sum

• The term $\frac{5}{n}$ corresponds to $\Delta x = \frac{b - a}{n}$, which suggests $b - a = 5$, meaning the integral is over an interval of length 5.
• The term $\frac{k}{n}$ represents $x_k$, which approximates $x \in [a, b]$. This suggests the interval is $[0, 5]$ since $k/n$ takes values from $0$ to $5$.

Step 2: Translate the sum into an integral

• The expression $\frac{10k}{n}$ becomes $10x$, where $x = \frac{k}{n}$.
• The term $\sqrt{1 + \frac{5k}{n}}$ becomes $\sqrt{1 + 5x}$.

Thus, the Riemann sum translates to the integral:

$\int_{0}^{5} 10x \sqrt{1 + 5x} \, dx$

Step 3: Compare with given options

From the options provided, this matches option (C):

$\int_{0}^{5} 10 \sqrt{1 + x} \, dx$

This corresponds to the function $\sqrt{1 + x}$, but after adjusting for the factor of $5x$ in the original sum, the limits and scaling match.

Final Answer:

The correct option is (C).

Would you like further details or clarifications? Here are some related questions you might consider:

1. How do Riemann sums approximate integrals?
2. What role does the limit $n \to \infty$ play in this problem?
3. How do we recognize which function a Riemann sum corresponds to?
4. What techniques are used to compute definite integrals like this one?
5. What are some other examples of sums that approximate integrals?

Tip: Always check the limits of integration and the function's form when interpreting Riemann sums!