The given problems involve expressing Riemann sums as definite integrals. Let’s rewrite each limit as a definite integral:

Problem 1:

$\lim_{\|P\|\to 0} \sum_{k=1}^n c_k^2 \Delta x_k, \text{ where } P \text{ is a partition of } [0, 2].$ This is a Riemann sum for the function $f(x) = x^2$ on the interval $[0, 2]$. The integral is: $\int_0^2 x^2 \, dx.$

Problem 2:

$\lim_{\|P\|\to 0} \sum_{k=1}^n 2c_k^3 \Delta x_k, \text{ where } P \text{ is a partition of } [-1, 0].$ This is a Riemann sum for the function $f(x) = 2x^3$ on the interval $[-1, 0]$. The integral is: $\int_{-1}^0 2x^3 \, dx.$

Problem 3:

$\lim_{\|P\|\to 0} \sum_{k=1}^n (c_k^2 - 3c_k) \Delta x_k, \text{ where } P \text{ is a partition of } [-7, 5].$ This is a Riemann sum for the function $f(x) = x^2 - 3x$ on the interval $[-7, 5]$. The integral is: $\int_{-7}^5 (x^2 - 3x) \, dx.$

Problem 4:

$\lim_{\|P\|\to 0} \sum_{k=1}^n \left(\frac{1}{c_k}\right) \Delta x_k, \text{ where } P \text{ is a partition of } [1, 4].$ This is a Riemann sum for the function $f(x) = \frac{1}{x}$ on the interval $[1, 4]$. The integral is: $\int_1^4 \frac{1}{x} \, dx.$

Problem 5:

$\lim_{\|P\|\to 0} \sum_{k=1}^n \frac{1}{1 - c_k} \Delta x_k, \text{ where } P \text{ is a partition of } [2, 3].$ This is a Riemann sum for the function $f(x) = \frac{1}{1 - x}$ on the interval $[2, 3]$. The integral is: $\int_2^3 \frac{1}{1 - x} \, dx.$

Would you like me to compute these integrals or provide further explanations?

Related Questions:

1. How do you evaluate the definite integral of a polynomial function like $x^2$?
2. What is the significance of the Riemann sum in defining the integral?
3. How do you deal with integrals of rational functions, such as $\frac{1}{x}$?
4. What happens if the function is not continuous in the interval of integration?
5. Can a Riemann sum converge to an improper integral if the interval is infinite?

Tip:

Always analyze the function in the Riemann sum to identify the corresponding integral’s limits and integrand.