https://file.aiquickdraw.com/mathbot/file/1726074753627xfs0equ9.jpg

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)**.

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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!

Which of the following definite integrals is equal to the limit sum given in the image?

Learn how to interpret Riemann sums as definite integrals with this detailed problem. It involves converting a limit sum into an integral, identifying the correct function, and solving it using key calculus concepts.

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