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The question asks to determine the function and region whose area is equal to the given limit:

\[
\lim_{n \to \infty} \sum_{i=1}^{n} \left( \frac{25}{n} \cdot \sqrt{4 + \left(\frac{i^2}{n^2}\right)} \right)
\]

This sum represents a Riemann sum, which approximates the area under a curve. From the sum's structure, we can interpret the corresponding function, region, and limits of integration.

### Key Insights:
1. The term \( \frac{25}{n} \) suggests that the total length of the interval is 25, implying an integration over the interval \([0, 5]\), since \( \frac{25}{n} \) is related to the width of each subinterval in a Riemann sum.
2. The expression \( \sqrt{4 + \frac{i^2}{n^2}} \) implies the function inside the integral is \( \sqrt{4 + x^2} \), where \( x = \frac{i}{n} \).
3. Therefore, the integral corresponds to:
\[
\int_0^5 \sqrt{4 + x^2} \, dx
\]

### Matching the Function and Region:
Looking at the answer options:

- **Option B**: 
  \[
  f(x) = \sqrt{4 + x^2}, \quad 0 \leq x \leq 5
  \]
  
This matches the form of the integral derived from the Riemann sum. Thus, **Option B** is correct.

Would you like further details on the process? Here are some additional questions to explore the concept:

1. How can we derive a Riemann sum from any given integral?
2. What are the steps to convert a limit of a sum into an integral?
3. How does the choice of the function in a Riemann sum affect the approximation of the area?
4. What are common functions that lead to integrals involving square roots, such as \( \sqrt{4 + x^2} \)?
5. What are the key differences between definite and indefinite integrals?

**Tip**: When working with Riemann sums, identifying the relationship between the sum and the corresponding integral is essential for understanding the limit.

Determine the function and region whose area is equal to the given limit.

This problem asks to determine the function and region whose area equals the given Riemann sum limit. The key concepts involve Riemann sums, definite integrals, and square root functions. Suitable for undergraduate students studying calculus.

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