The given question asks us to interpret the limit:

$\lim_{\Delta x \to 0} \sum_{i=1}^n 4c_i (7-c_i)^2 \Delta x_i$

as a definite integral on the interval $[0,8]$, where $c_i$ is any point in the $i$-th subinterval.

Step-by-step Solution:

1. Identify the form of the sum: The given expression represents a Riemann sum, which approximates the value of a definite integral. The general Riemann sum is of the form: $\sum_{i=1}^n f(c_i) \Delta x_i$ Here, $f(c_i)$ is the function value at some point $c_i$ within the $i$-th subinterval.

2. Determine the function $f(x)$: In this case, $f(c_i) = 4c_i(7-c_i)^2$. So the function we are integrating is: $f(x) = 4x(7-x)^2$

3. Set the interval: The interval of integration is given as $[0,8]$.

4. Write the definite integral: The limit of the Riemann sum as $\Delta x \to 0$ becomes the definite integral: $\int_0^8 4x(7-x)^2 \, dx$

Correct Answer:

The correct choice is:

$\int_0^8 4x(7-x)^2 \, dx$

Would you like more details on how Riemann sums convert into definite integrals or any other aspect of this question?

Related Questions:

1. What is the meaning of a Riemann sum, and how does it approximate integrals?
2. How would the integral change if the limits of integration were $[1,5]$ instead of $[0,8]$?
3. What is the expanded form of the function $4x(7-x)^2$ before integration?
4. Can this integral be solved analytically, and if so, what is its value?
5. How do different choices of $c_i$ within the subinterval affect the Riemann sum?

Tip:

When interpreting Riemann sums, always identify the function being evaluated and the interval of integration to write the integral correctly!