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The image contains a limit expression related to the total mass of a thin metal rod. Here's a breakdown of the problem:

The limit is given as:

\[
\lim_{n \to \infty} \sum_{i=1}^{n} \sqrt{2 + \left(\frac{10i}{n}\right)^2} \Delta x
\]

- The length of the rod is \(10\) cm.
- The rod is divided into \(n\) segments of length \(\Delta x = \frac{10}{n}\).
- \(x_i\) is the point in the \(i\)-th segment.
- The mass is calculated as the sum over these segments.

The task is to express the limit as a definite integral involving the mass density function \(f(x)\).

### Step-by-step interpretation:

1. The given sum represents a Riemann sum for the mass of the rod as \(n \to \infty\).
2. The expression for the mass per segment includes the density function, and it can be rewritten in integral form.

The Riemann sum corresponds to the integral of the density function \(f(x)\) over the length of the rod. 

We need to identify \(a\), \(b\), and \(f(x)\), where the integral has the form:

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

- The rod extends from \(x = 0\) to \(x = 10\), so the limits of integration are \(a = 0\) and \(b = 10\).
- From the sum, we can identify the function inside the sum as related to the density function \(f(x)\).

Thus, the density function is:

\[
f(x) = \sqrt{2 + x^2}
\]

### Conclusion:
The limit is expressed as the definite integral:

\[
\int_0^{10} \sqrt{2 + x^2} \, dx
\]

The values are:
- \(a = 0\)
- \(b = 10\)
- \(f(x) = \sqrt{2 + x^2}\)

Would you like further details or explanations?

### Related Questions:
1. How is a Riemann sum used to approximate integrals?
2. What is the interpretation of the density function in this context?
3. How can you calculate the definite integral of \( \sqrt{2 + x^2} \)?
4. What other physical problems can be modeled using similar Riemann sums?
5. How does the density function affect the total mass distribution along the rod?

**Tip:** When transforming a Riemann sum to an integral, observe how the sum's terms relate to the function you are integrating over the interval.

Suppose the limit \( \lim_{n \to \infty} \sum_{i=1}^{n} \sqrt{2 + \left( \frac{10i}{n} \right)^2 } \Delta x \) represents the total mass of a thin metal rod with length 10 cm where the rod has been divided into n segments of length \( \Delta x \), and \( x_i \) is a point in the i-th segment of the rod. If \( f(x) \) is the rod's density function, where x is the distance from one end of the rod, express this limit as a definite integral. Identify a, b, and f(x).

Solve the problem of finding the total mass of a thin metal rod using a Riemann sum. Convert the sum to a definite integral involving the rod's density function. Understand the steps to express the sum as an integral and identify the density function, limits of integration, and related mathematical concepts.

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