To solve this problem and express the given limit as a definite integral, let's break it down step by step.

Problem Breakdown

The given expression is:

$\lim_{\|P\| \to 0} \sum_{k=1}^{n} c_k^3 \Delta x_k,$

where $P$ is a partition of the interval $[2, 8]$. Here:

• $c_k$ is a sample point in the subinterval.
• $\Delta x_k$ is the width of the subinterval $[x_{k-1}, x_k]$.
• $\|P\|$ denotes the norm of the partition, which approaches 0 as $n \to \infty$.

Expressing as a Definite Integral

This sum is a Riemann sum representation of the integral:

$\int_{a}^{b} f(x) \, dx.$

In our case:

• The interval is $[2, 8]$.
• The function being integrated is $f(x) = x^3$ because $c_k^3$ approximates $x^3$ in the subinterval as $n \to \infty$.

Thus, the limit can be expressed as the following definite integral:

$\int_{2}^{8} x^3 \, dx.$

Evaluating the Integral

Now, let's compute the integral:

$\int_{2}^{8} x^3 \, dx = \left[\frac{x^4}{4}\right]_{2}^{8}.$

Substitute the bounds:

$\left(\frac{8^4}{4}\right) - \left(\frac{2^4}{4}\right) = \left(\frac{4096}{4}\right) - \left(\frac{16}{4}\right) = 1024 - 4 = 1020.$

Final Answer

The limit is equivalent to the value of the definite integral:

$\int_{2}^{8} x^3 \, dx = 1020.$

Would you like a deeper explanation of Riemann sums or additional examples? Here are related questions to consider:

1. How would the expression change if the function were $x^2$ instead of $x^3$?
2. What happens if we change the interval from $[2, 8]$ to $[a, b]$?
3. How do we interpret the Riemann sum geometrically?
4. What if the partition $P$ was uniform? How would it simplify the sum?
5. How can this concept be extended to multi-variable calculus?

Tip: Understanding Riemann sums is crucial because they provide the foundation for defining definite integrals as limits of sums.