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.