To rewrite the given limit as a definite integral, observe the structure of the Riemann sum. The limit provided is:

$\lim_{\|\Delta\|\to 0} \sum_{i=1}^n (2c_i + 10) \Delta x_i,$

where $c_i$ is any point in the $i$th subinterval, and the interval of integration is $[-4, 8]$. This is the form of a Riemann sum for a definite integral.

The general form of a Riemann sum for a definite integral is:

$\int_a^b f(x) \, dx = \lim_{\|\Delta\|\to 0} \sum_{i=1}^n f(c_i) \Delta x_i,$

where $a$ and $b$ are the limits of integration, $f(x)$ is the integrand, and $\Delta x_i$ is the width of each subinterval.

Here:

• The interval $[-4, 8]$ gives the limits of integration $a = -4$ and $b = 8$.
• The function being integrated is $f(x) = 2x + 10$.

Thus, the definite integral corresponding to the given Riemann sum is:

$\int_{-4}^8 (2x + 10) \, dx.$

Would you like me to solve this integral or provide a detailed explanation of the steps?

Related Questions:

1. What are Riemann sums, and how do they approximate definite integrals?
2. Why does taking the limit of the Riemann sum as $\|\Delta\|\to 0$ yield the exact value of the integral?
3. What does the function $2x + 10$ represent in the context of this integral?
4. How would the result change if the limits of integration were different?
5. Can you provide a geometric interpretation of this definite integral?

Tip:

When working with Riemann sums, always identify the function $f(x)$ and the interval $[a, b]$ first, as this directly maps the sum to an integral.