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To solve this problem, let’s analyze and interpret each part:

### Problem Breakdown
The problem is about determining the integral \(\int_a^b f(x) \, dx\) from a limit expression. You are given:
\[
\lim_{n \to \infty} \sum_{i=1}^n \frac{2i}{n^2} \left(2 + \frac{2i}{n}\right)
\]
This expression resembles a Riemann sum that approximates an integral of the form \(\int_a^b f(x) \, dx\).

### Part (a): Determining \(\Delta x\), \(x_i\), \(a\), \(b\), and \(f(x)\)
To express the sum as a Riemann sum, let's interpret each part of the sum:

1. **Riemann Sum Structure**: A Riemann sum for \(\int_a^b f(x) \, dx\) with \(n\) subintervals has the form:
   \[
   \sum_{i=1}^n f(x_i) \Delta x
   \]
   where:
   - \(\Delta x = \frac{b - a}{n}\),
   - \(x_i = a + i \Delta x\),
   - \(f(x_i)\) represents the function evaluated at \(x_i\).

2. **Identifying \(\Delta x\)**: By comparing the sum,
   \[
   \frac{2i}{n^2} \left(2 + \frac{2i}{n}\right),
   \]
   we can see that \(\frac{2}{n}\) likely represents \(\Delta x\). Hence, \(\Delta x = \frac{2}{n}\).

3. **Identifying \(x_i\)**: Since \(\Delta x = \frac{2}{n}\), we can rewrite \(x_i\) as:
   \[
   x_i = a + i \Delta x.
   \]
   By inspecting the expression \(2 + \frac{2i}{n}\) in the sum, we observe that:
   \[
   x_i = 2 + \frac{2i}{n}.
   \]
   This implies \(a = 2\) and \(b = 4\) (as \(x_i\) goes from \(2\) to \(4\) when \(i\) goes from \(1\) to \(n\)).

4. **Identifying \(f(x)\)**: The function \(f(x)\) should match the remaining part of the sum that isn’t part of \(\Delta x\). This means:
   \[
   f(x) = x.
   \]

### Part (a) Solution Summary
Thus, we have:
- \(\Delta x = \frac{2}{n}\),
- \(x_i = 2 + \frac{2i}{n}\),
- \(a = 2\),
- \(b = 4\),
- \(f(x) = x\).

### Part (b): Calculating the Value of the Limit
The expression given is:
\[
\lim_{n \to \infty} \sum_{i=1}^n \frac{2i}{n^2} \left(2 + \frac{2i}{n}\right).
\]
We can rewrite this as a Riemann sum for the integral:
\[
\int_2^4 x \, dx.
\]
To calculate this integral:
\[
\int_2^4 x \, dx = \left[ \frac{x^2}{2} \right]_2^4 = \frac{4^2}{2} - \frac{2^2}{2} = \frac{16}{2} - \frac{4}{2} = 8 - 2 = 6.
\]

### Final Answer
For part (a):
\[
(\Delta x, x_i, a, b, f(x)) = \left(\frac{2}{n}, 2 + \frac{2i}{n}, 2, 4, x\right).
\]

For part (b):
\[
\lim_{n \to \infty} \sum_{i=1}^n \frac{2i}{n^2} \left(2 + \frac{2i}{n}\right) = 6.
\]

Would you like further details or have any questions?

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Here are five related questions for further exploration:
1. What is the purpose of using Riemann sums in approximating definite integrals?
2. How do different values of \(a\) and \(b\) affect the integral's value?
3. How would this problem change if \(f(x)\) were a different function, such as \(f(x) = x^2\)?
4. Why does the limit as \(n \to \infty\) give the exact value of the integral?
5. How would you interpret the meaning of \(\Delta x\) and \(x_i\) in practical terms?

**Tip**: Riemann sums become closer to the true integral value as the number of intervals \(n\) increases, as the approximation error decreases with finer partitions.

The following limit represents the definite integral \(\int_a^b f(x) \, dx\) for some real numbers \(a\) and \(b\), and an integrable function \(f\): 
\[ \lim_{n \to \infty} \sum_{i=1}^n \frac{2i}{n^2} \left(2 + \frac{2i}{n}\right) \]
Find \(\Delta x\) and \(x_i\) from this expression and deduce the values of \(a\) and \(b\), and the function \(f\). Then compute the value of the limit.

Explore how to calculate a definite integral using a Riemann sum and a limit. This problem involves finding \(\Delta x\), \(x_i\), and deducing values for \(a\) and \(b\). Learn to interpret the limit \(\lim_{n \to \infty} \sum_{i=1}^n \frac{2i}{n^2} (2 + \frac{2i}{n})\) as an integral. Suitable for undergraduate students studying calculus.

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