1. **Problem:** Express the limit \(\lim_{\|P\|\to 0} \sum_{k=1}^n c_k^2 \Delta x_k\) as a definite integral, where \(P\) is a partition of \([0, 2]\). 2. **Formula and Explanation:** The limit of a Riemann sum \(\lim_{\|P\|\to 0} \sum_{k=1}^n f(c_k) \Delta x_k\) represents the definite integral of the function \(f(x)\) over the interval \([a,b]\). 3. **Identify the function and interval:** Here, \(f(x) = x^2\) and the interval is \([0, 2]\). 4. **Write the integral:** $$\int_0^2 x^2 \, dx$$ 5. **Evaluate the integral:** $$\int_0^2 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^2 = \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3}$$ 6. **Interpretation:** The limit of the sum is the area under the curve \(y = x^2\) from 0 to 2, which equals \(\frac{8}{3}\). **Final answer:** $$\lim_{\|P\|\to 0} \sum_{k=1}^n c_k^2 \Delta x_k = \int_0^2 x^2 \, dx = \frac{8}{3}$$